IEEE Transaction on computers (January)

101 Pages·2005·8.54 MB·English

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IEEETRANSACTIONSONCOMPUTERS, VOL.54, NO.1, JANUARY2005 1
Error-Detection Codes:
Algorithms and Fast Implementation
Gam D. Nguyen
Abstract—BinaryCRCsareveryeffectiveforerrordetection,buttheirsoftwareimplementationisnotveryefficient.Thus,manybinary
non-CRCcodes(whicharenotasstrongasCRCs,butcanbemoreefficientlyimplementedinsoftware)areproposedasalternatives
toCRCs.Thenon-CRCcodesincludeWSC,CXOR,one’s-complementchecksum,Fletcherchecksum,andblock-paritycode.Inthis
paper,wepresentageneralalgorithmforconstructingafamilyofbinaryerror-detectioncodes.Thisfamilyislargebecauseitcontains
allthesenon-CRCcodes,CRCs,perfectcodes,aswellasotherlinearandnonlinearcodes.Inadditiontounifyingtheseapparently
disparatecodes,ouralgorithmalsogeneratessomenon-CRCcodesthathaveminimumdistance4(likeCRCs)andefficientsoftware
implementation.
IndexTerms—Fasterror-detectioncode,Hammingcode,CRC,checksum.
(cid:1)
1 INTRODUCTION
EFFICIENT implementation of reliable error-protection performance comparisons of CRCs and these non-CRC
algorithms plays a vital role in digital communication codes. Perhaps the key reason for the appearance of the
and storage because channel noise and system malfunc- non-CRC codes is that CRCs are not very efficiently
tion introduce errors in received and retrieved messages. implemented in software. Software complexity refers to
Here, we focus on binary error-detection codes that have the number of programming operations and hardware
low overhead and minimum distance d(cid:1)4. Popular complexity refers to the number of gates required for code
error-detection codes used in practice include CRCs that implementation. Investigations reported in [4], [9] indicate
are generated by binary polynomials such as X16þX15þ thatsoftware processingof CRCs is slowerthan that of the
X2þ1 (called CRC-16) and X16þX12þX5þ1 (called non-CRC codes. Thus, it is desirable to design error-
CRC-CCITT). detection codes that are reliable and of low complexity.
Anh-bitCRCgeneratedbyGðXÞ¼ðXþ1ÞPðXÞ,where One code is better than another if, for a fixed number of
PðXÞ is a primitive polynomial of degree h(cid:2)1, has the checkbitsh,ithaslargerminimumdistanced,largerburst-
following desirable properties [1]. The CRC has maximum error-detecting capability b, longer maximum codeword
codeword length of 2h(cid:2)1(cid:2)1 bits and minimum distance length lmax, and lower complexity.
d¼4,i.e.,alldoubleandodderrorsaredetected.Thiscode An important performance measure of a code, which is
also detects any error burst of length h bits or less, i.e., its notaddressedinthispaper,isitsprobabilityofundetected
burst-error-detecting capability is b¼h. The guaranteed error.Forthebinarysymmetricchannel,thisprobabilitycan
error-detectioncapabilityofthish-bitCRCisnearlyoptimal beexpressedintermsoftheweightdistributionofthecode.
because its maximum codeword length almost meets the In general, the problem of computing the probability of
upper bound 2h(cid:2)1 and its burst-error-detecting capability undetected error is NP-hard [7]. Some methods for
meets the upper bound h. The CRC is also efficiently calculating or estimating this probability are given in [7].
implemented by special-purpose shift-register hardware. Becausetheminimumdistancedisoftenconsidered the
Although CRCs have nearly optimal properties and most important parameter, Fig. 1 ranks CRC as the best
efficient hardware implementation, many binary non-CRC code, WSC the second best, and so on. Although the WSC,
codes are proposed as alternatives to CRCs. These codes, Fletcherchecksum,andCXORaredefinedonlyforaneven
developed over many years and often considered as numberofcheckbitsh,bothevenandoddhcanbeusedfor
unrelated to each other, do not have the CRC’s desirable theothercodes.TheCRC,WSC,andFletcherchecksumcan
properties. Such non-CRC codes include weighted sum be extended to have infinite length, but their minimum
code (WSC), Fletcher checksum (used in ISO), one’s- distances all reduce to 2. Some discussions of burst-error-
complement checksum (used in Internet), circular-shift detecting capability b are given in Appendix C (which can
andexclusive-ORchecksum(CXOR),andblock-paritycode befoundontheComputerSocietyDigitalLibraryathttp://
(Fig. 1). See [4], [5], [9], [14] for implementation and computer.org/tc/archives.htm). In this paper, we focus on
code implementation by means of software. Because
computers can process information in blocks of bits (e.g.,
. TheauthoriswiththeInformationTechnologyDivision,NavalResearch bytesorwords),codeshavingefficientsoftwareimplemen-
Laboratory,Washington,DC20375.E-mail:[email protected].
tationshouldalsobe processedin blocksof bits.Then,itis
Manuscript received 27 Mar. 2003; revised 26 Feb. 2004; accepted 30 July natural to express code lengths in terms of the number of
2004;publishedonline16Nov.2004.
blocksnandeachblockissbits,i.e.,thetotalnumberofbits
For information on obtaining reprints of this article, please send e-mail to:
[email protected],andreferenceIEEECSLogNumber118495. is ns. Most modern processors can efficiently handle block
U.S.GovernmentWorkNotProtectedbyU.S.Copyright
TLFeBOOK
2 IEEETRANSACTIONSONCOMPUTERS, VOL.54, NO.1, JANUARY2005
denotes a systematic code with l¼ code length, k¼
information block length, and d¼ minimum distance.
Finally,ifY1 andY2 arem1-tupleandm2-tuple,respectively,
thenY ¼ðY1;Y2Þ denotes the concatenation of Y2 to Y1, i.e.,
Y is an ðm1þm2Þ-tuple. Note thatY canalso be written as
Y ¼Y1Xm2 þY2. For ease of cross-referencing, we usually
label blocks of text as “Remarks.” These remarks are
integral parts of our presentation and they should not be
viewed as isolated observations or comments.
Fig. 1. Error-detection capabilities of binary codes (d¼ minimum 2 A GENERAL ALGORITHM FOR ERROR-DETECTION
distance, b¼ burst-error-detecting capability, h¼ number of check CODES
bits,lmax¼ maximumcodewordlength).
In this section, we define a binary code so that each of its
size s¼8;16;32;64 bits. General-purpose computers and codewords consists of n tuples Q0;Q1;...;Qn(cid:2)1, each tuple
compilersareincreasinglyfasterandbetter.Thus,software is s bits. This code is not necessarily systematic and is
algorithms become more relevant and desirable. Software formulated abstractly to facilitate the development of its
algorithms are increasingly used in operations, modeling, mathematical properties. For practical use, we then con-
simulations, and performance analysis of systems and struct systematic versions of the code. Fast software
networks. An important advantage of software implemen- versions of the code will be presented later in Section 4. It
tation is its flexibility: It is much simpler to modify a isimportanttonotethatQiisanuppercaseletter,so,byour
software program than to modify a chip full of hardwired conventions, Qi is a polynomial of the variable X, i.e.,
gates and buses.
Qi ¼QiðXÞ. Further, being an s-tuple, Qi is also a
In this paper, we present a general algorithm and its
polynomial of degree less than s. The polynomial notation
systematicversionsforconstructingalargefamilyofbinary
facilitates the mathematical developments of codes. The
error-detectioncodes(Section2).Thisfamilycontainsallthe
tuple notation is more appropriate for software implemen-
codesinFig.1andotherlinearandnonlinearcodesforerror
tationofcodesbecauseans-tupleisagroupofsbits,which
detection. We unify the treatment of these seemingly
can be easily processed by computers. Note also that the
unrelated codes by showing that CRCs and the non-CRC
codesallcomefromasinglealgorithm(Section3).Further, ns-tuple ðQ0;Q1;...;Qn(cid:2)2;Qn(cid:2)1Þ is equivalent to the poly-
thealgorithmcanproducesomenon-CRCcodesthatarenot nomial Pni¼(cid:2)01QiXðn(cid:2)1(cid:2)iÞs of degree less than ns.
onlyreliable(i.e.,havingminimumdistance4asCRCs),but First,letC1 beabinarycodewithlengthsandminimum
alsohavefastsoftwareimplementation(Section4).Wethen distanced1.Letrandnbeintegerssuchthat1(cid:1)n(cid:1)2r.Let
summarizeandconcludethepaper(Section5).Thepaperis W0;W1;...;Wn(cid:2)1bedistinctpolynomialsofdegreelessthan
supplementedwithappendices(whichcanbefoundonthe r.LetM beapolynomialofdegreersuchthatM andXare
Computer Society Digital Library at http://computer.org/ relatively prime, i.e., gcdðM;XÞ¼1. Also, Qi is an s-tuple,
tc/archives.htm)thatincludetheoremproofs,codesegments i(cid:3)0. Now, we are ready to introduce a new code that is
implemented in C programming language, as well as simplycalled“thecodeC”andisthefocusofthispaper.
discussions of CRCs, WSCs, and CXORs. The preliminary Algorithm1.LetC be thebinarycodesuchthateachofits
versionofthispaperispresentedin[10]. codewords
1.1 Notations and Conventions
ðQ ;Q ;...;Q ;Q Þ ð1Þ
0 1 n(cid:2)2 n(cid:2)1
We consider polynomials over only the binary field GF(2),
satisfies the following two conditions:
i.e.,thepolynomialoperationsareperformedinpolynomial
arithmetic modulo 2. Let A¼AðXÞ and B¼BðXÞ be two !
n(cid:2)1
X
polynomials, then AmodB is the remainder polynomial QW modM ¼0 ð2Þ
i i
that is obtained when A is divided by B with i¼0
degðAmodBÞ<degðBÞ. To ease the presentation of many
different codes (which can result in a large number of n(cid:2)1
X
parameters),weadoptthefollowingsweepingconventions. Qi 2C1: ð3Þ
A j-tuple ða0;...;aj(cid:2)2;aj(cid:2)1Þ denotes the binary polynomial i¼0
a0Xj(cid:2)1þ...þaj(cid:2)2Xþaj(cid:2)1 of degree less than j. In this Remark 1.
paper, lowercase letters (such as h and a0) denote
nonnegative integers. The letters C and C1 denote codes, 1. C is nonlinear if C1 is nonlinear.
other uppercase letters (such as A and Qi) denote 2. From (3), the codewords of C have even weights
polynomials (or tuples), and X denotes the variable (or if the codewords of C have even weights.
1
indeterminate)ofthesepolynomials.Further,thevariableX 3. ThecodeC1inAlgorithm1canbenonsystematic.
will be omitted from all polynomials, i.e., AðXÞ will be However, we focus only on systematic codes,
denoted as A. We denote ui as the i-tuple whose which are more often used in practice. Thus, we
components are all us, u2f0;1g. The notation ðl;k;dÞ assumethatC1 isan(s;s(cid:2)m;d1Þsystematiccode
TLFeBOOK
NGUYEN: ERROR-DETECTIONCODES:ALGORITHMSANDFASTIMPLEMENTATION 3
with m check bits, 0(cid:1)m(cid:1)s. Let F be the Y ¼Q0þQ1þðU1X2þP1ÞþU2X3
encoder of C1. Then, each codeword of C1 is ¼Q0þQ1þQ2þU2X3;
UXmþFðUÞ¼ðU;FðUÞÞ,whereU isaninforma-
tion ðs(cid:2)mÞ-tuple and FðUÞ is the corresponding which is a 4-tuple. Thus, Y can be written as
check m-tuple. Y ¼Y1X3þY2 ¼ðY1;Y2Þ, where Y1 is a 1-tuple and Y2
4. InAlgorithm1,theweightsW0;W1;...;Wn(cid:2)1 can is a 3-tuple. Finally, we compute P2 ¼Y2þðY1;Y1;Y1Þ,
which is a 3-tuple.
be chosen to be distinct polynomials of degree
less than r because 1(cid:1)n(cid:1)2r. However, Algo- Now, we will show that the codeword
rithm 1 can be extended to allow n>2r, then ðQ0;Q1;Q2;Q3Þ¼ðQ0;Q1;U1;P1;U2;P2Þ satisfies (2) and
SWe0c;tiWon1;3..l.a;tWern)(cid:2).1 will not always be distinct (see (P13¼)ðQ0Wi n0þQ1AWl1þgUo1Xr2iÞtmhodmM, we1h.aveS i n c e
5. All the codes considered in this paper are binary
0¼ðQ W þQ W þU X2þP ÞmodM:
0 0 1 1 1 1
codes,i.e.,theircodewordsconsistofdigits0or1.
In particular, the code C is a binary code whose Then, 0¼ðQ0W0þQ1W1þQ2W2þQ3W3ÞmodM be-
codewords are ns bits. Computers can efficiently cause Q2W2 ¼U1X2þP1 and Q3W3 ¼0. Thus,
process groups of bits. Thus, as seen in (1), each ðQ0;Q1;Q2;Q3Þ satisfies (2). Next,
ns-bit codeword is grouped into n tuples, s bits
each. Note that this binary code C can also be Q0þQ1þQ2þQ3 ¼Y þU2X3þQ3
viewed as a code in GFð2sÞ, i.e., as a code whose ðbecauseY ¼Q þQ þQ þU X3Þ
0 1 2 2
codewords consist of n symbols, each symbol
¼Y þU X3þðU ;P Þ
belongs to GFð2sÞ. More generally, suppose that 2 2 2
ns¼xy for some positive integers x and y, then ¼Y þP2 ½becauseðU2;P2Þ¼U2X3þP2(cid:4)
this same binary code C can also be viewed as a ¼ðY ;Y ÞþY þðY ;Y ;Y Þ
1 2 2 1 1 1
codewhosecodewordsconsistofxsymbols,each ¼ðY ;Y ;Y ;Y Þ2C :
symbol belongs to GFð2yÞ. In the extreme case 1 1 1 1 1
(x¼1, y¼ns), the code C is also a code whose Thus, ðQ0;Q1;Q2;Q3Þ¼ðQ0;Q1;U1;P1;U2;P2Þ also satis-
codewords consist of only one symbol that fies(3).ByexchangingP1andU2,thecodewordbecomes
belongs to GFð2nsÞ. Note that, when the same ðQ0;Q1;U1;U2;P1;P2Þ, which is a codeword of a sys-
code is viewed in different alphabets, their tematic codebecauseðQ0;Q1;U1;U2Þarethe 11 informa-
respectiveminimumdistancescanbeverydiffer- tion bits and ðP1;P2Þ are the 5 corresponding check bits.
ent. For example, consider the binary repetition Because d1 ¼4, dC ¼4 by Theorem 1.2. Thus, C is
codef0k;1kgoflengthk>1.Whenviewedasthe identical to the ð16;11;4Þ extended Hamming code.
binary code over GF(2), this code has minimum
distance d¼k. But, when viewed in GFð2kÞ, this 2.1 Systematic Encoding
same code has minimum distance d¼1. In general, the binary code C in Algorithm 1 is not
Let d1 and dC be the minimum distances of the binary systematic. Now, we construct its systematic versions.
codesC1 andC,respectively,inAlgorithm1.Wethenhave RecallthatristhedegreeofthemodulatingpolynomialM
thefollowingtheoremthatisproveninAppendixA(which and s is the number of bits contained in each tuple Qi. Let
can be found on the Computer Society Digital Library at r(cid:1)s and suppose that information tuples
http://computer.org/tc/archives.htm).
ðQ ;Q ;...;Q ;U ;U Þ ð4Þ
0 1 n(cid:2)3 1 2
Theorem 1.
are given, where U1 is an ðs(cid:2)rÞ-tuple and U2 is an
1. dC (cid:3)3 if d1 (cid:3)3. ðs(cid:2)mÞ-tuple. We wish to append a check r-tuple P1 and
2. dC ¼4 if d1 (cid:3)4. a check m-tuple P2 to (4) so that the resulting codeword is
Example1.Now, we illustrate Algorithm 1 by construct- ðQ ;Q ;...;Q ;U ;U ;P ;P Þ: ð5Þ
0 1 n(cid:2)3 1 2 1 2
ing a simple binary code C. Let s¼4, m¼3, r¼2,
and n¼2r ¼4. Thus, each codeword of the code C Thus, the code C is ns bits long and has h¼rþm check
is a 16-tuple ðQ0;Q1;Q2;Q3Þ, where each Qi is a bits. Denote dC as its minimum distance, then C is an
4-tuple. Let M ¼X2þXþ1 be the modulating ðns;ns(cid:2)r(cid:2)m;dCÞ code. Then, we have the following
algorithm that is proven in Appendix A (which can be
(primitive) polynomial. Let the weighting polynomials
in (2) be W0 ¼Xþ1, W1 ¼X, W2 ¼1, and W3 ¼0. found on the Computer Society Digital Library at http://
Let C1 ¼fð0;0;0;0Þ;ð1;1;1;1Þg, i.e., C1 is the ð4;1;4Þ computer.org/tc/archives.htm).
repetition code. Now, we wish to specify the desired Algorithm 1a. When r(cid:1)s, the two check tuples of a
codeword ðQ0;Q1;Q2;Q3Þ. Let Q0 and Q1 be two systematicversionofthebinarycodeCcanbecomputedby
arbitrary 4-tuples. Then, Q2 and Q3 are determined as !
n(cid:2)3
follows: Let U1 and U2 be arbitrary 2-tuple and 1-tuple, P ¼ XQW þU Xr modM ð6Þ
respectively. Then, we define Q2 ¼ðU1;P1Þ and 1 i¼0 i i 1
Q3 ¼ðU2;P2Þ,whereP1andP2aredeterminedasfollows:
F i r s t , c o mp u te t h e c h e c k 2- t up l e
P ¼Y þFðY Þ; ð7Þ
P1 ¼ðQ0W0þQ1W1þU1X2ÞmodM.Next,define 2 2 1
TLFeBOOK
4 IEEETRANSACTIONSONCOMPUTERS, VOL.54, NO.1, JANUARY2005
where Wi 6¼0;1 and F is the encoder of C1 as defined in
Remark 1.3. The tuples Y and Y are determined as
1 2
follows: Let
n(cid:2)3
X
Y ¼ Q þU XrþP þU Xm;
i 1 1 2
i¼0
which is an s-tuple that can be written as
Y ¼Y1XmþY2 ¼ðY1;Y2Þ, where Y1 and Y2 are an ðs(cid:2)
mÞ-tuple and an m-tuple, respectively.
Remark 2. After P1 is computed, P2 is easily computed
whenC1 isoneofthefollowingfourtypesofcodes:The
firsttwotypesofcodes,givenin1and2below,arevery
trivial,buttheyareusedlaterinSection3toconstructall Fig.2.ConstructionofthecodesC using16checkbits.
thecodesinFig.1.Thenexttwotypesofcodes,givenin
3 and 4 below, are commonly used in practice for error For example, let s¼8, then a¼1 and b¼4 if r¼12,
control. whereas a¼1 and b¼8 if r¼16. Thus, P1 can be
stored in aþ1 tuples: The first tuple is b bits and
1. If m¼s, then C1 ¼f0sg, which is an ðs;0;d1Þ each of the next a tuples is s bits. Now, assume that
cfiondeed,.wThheisrevethrye mtriivniiamlucmodediisstacnacleledd1aisuusenldeses- information tuples ðQ0;Q1;...;Qn(cid:2)a(cid:2)3;U1;U2Þ are given,
code because it carries no useful information. whereeachQi issbits,U1 iss(cid:2)bbits,andU2 iss(cid:2)mbits.
We assume here that n(cid:2)a(cid:2)3(cid:3)0 or n(cid:3)aþ3, to avoid
However, itcandetectanynumber oferrors, i.e.,
Fwuertchaenr,asitsigcnand1b¼e 1shofowrnthtihsaptarTthiceuolraermcod1.e2. tðrQiv0;iaQli1t;y...W.;eQwn(cid:2)ias(cid:2)h3t;oUa1;pUp2eÞnsdotwthoatcheckstuplesP1andP2to
remains valid when m¼s, i.e., dC ¼4 if ðQ0;Q1;...;Qn(cid:2)a(cid:2)3;U1;U2;P1;P2Þ
C1 ¼f0sg. Then, from Algorithm 1a, we have
U2 ¼0, F ¼0s, Y1 ¼0, and becomes a codeword of a systematic ðns;ns(cid:2)r(cid:2)m;dCÞ
code.Then,wehavethefollowingalgorithmthatisprovenin
Xn(cid:2)3 AppendixA(whichcanbefoundontheComputerSociety
P ¼Y ¼Y ¼ Q þU XrþP :
2 2 i 1 1 DigitalLibraryathttp://computer.org/tc/archives.htm).
i¼0
Algorithm 1b. When r>s, the two check tuples of a
2. If m¼0, then C1 ¼f0;1gs, which is an ðs;s;1Þ systematicversionofthebinarycodeCcanbecomputedby
code. This very trivial code is called a powerless n(cid:2)a(cid:2)3 !
X
code because it protects no information. From P1 ¼ QiWiþU1Xr modM andP2 ¼Y2þFðY1Þ;
Algorithm 1a, we have Y2 ¼0, F ¼0, i¼0
n(cid:2)3 where F is the encoder of C1 and
X
Y ¼Y ¼ Q þU XrþP þU ;
1 i 1 1 2
W 6¼Xas;Xða(cid:2)1Þs;...;Xs;1;0:
i¼0 i
and P2 ¼0. The tuples Y1 and Y2 are determined as follows: Define
3. IfC1 isasystematiclinearcodewithparitycheck ! !
n(cid:2)a(cid:2)3 a
matrix H1 ¼½AI(cid:4), where A is an m(cid:5)ðs(cid:2)mÞ Y ¼ X Q þ(cid:1)U XbþP (cid:2)þ XP þU Xm;
matrix and I is the m(cid:5)m identity matrix, then i 1 10 1i 2
i¼0 i¼1
FðUÞ¼UAtr, where “tr” denotes matrix trans-
pose.Thus,P2 ¼Y2þFðY1Þ¼Y2þY1Atr ¼YHt1r. where P10 is a b-tuple and P11;...;P1a are s-tuples that
4. If C1 is a CRC generated by a polynomial M1 of satisfyP1 ¼ðP10;P11;...;P1aÞ.Then,Y isans-tuplethatcan
degree m, then FðUÞ¼ðUXmÞmodM1 (see Ap- bewrittenasY ¼Y1XmþY2 ¼ðY1;Y2Þ,whereY1andY2are
pendix B, which can be found on the Computer an ðs(cid:2)mÞ-tuple and an m-tuple, respectively.
Society Digital Library at http://computer.org/ Example2.RecallthatC isanðns;ns(cid:2)r(cid:2)m;dCÞcodethat
tc/archives.htm). Thus, is constructed by either Algorithm 1a (if r(cid:1)s) or
Algorithm 1b (if r>s). This code has h¼rþm check
P ¼Y þðY XmÞmodM ¼ðY XmþY ÞmodM
2 2 1 1 1 2 1 bits.Inthisexample,weassumethath¼16bitsandwe
¼Y modM1: present different ways to construct the codes C. The
results are shown in Fig. 2. For example, using
Algorithm 1a is for the case r(cid:1)s, where the check Algorithm 1b, we can construct the code C with the
r-tuple P1 can be stored in a single s-tuple. Now, we following parameters: s¼8, r¼12, m¼4, C1 ¼ð8;4;4Þ
consider the case r>s. Then, several s-tuples are code, a¼1, and b¼4 (a and b are not needed in
needed to store the check r-tuple P1. Because r>s, Algorithm 1a). Assume that the number of s-tuples
we can write r¼asþb, where a(cid:3)1 and 0<b(cid:1)s. satisfiesn(cid:1)2r,i.e.,thenumberofbitsineachcodeword
TLFeBOOK
NGUYEN: ERROR-DETECTIONCODES:ALGORITHMSANDFASTIMPLEMENTATION 5
is ns(cid:1)2rs¼215. Then, the weighting polynomials Wi P ¼Xn(cid:2)3QW modM and P ¼Xn(cid:2)3Q þP : ð8Þ
1 i i 2 i 1
can be chosen to be distinct. From Remark 2.1, we have
i¼0 i¼0
d1 (cid:3)4.Then,fromTheorem1.2,allthecodesC inFig.2
have minimum distance dC ¼4. Thus,thisspecialcodeC istheWSCpresentedin[4],[9].It
is shown in [3] that the WSC, when viewed as a code in
GFð2sÞ,isequivalenttoalengthenedsingle-errorcorrecting
3 SOME SPECIAL ERROR-DETECTION CODES
Reed Solomon code (see also [8, p. 323]).
ThissectionshowsthatthebinarycodeC ofAlgorithm1is
3.3 Block-Parity Code
generalinthesensethatitincludesallthecodesinFig.1and
other codes as special cases. Recall that Algorithm 1’s Suppose that r¼0 and m¼s. Thus, by Remark 2.1,
systematic version is either Algorithm 1a (if r(cid:1)s) or C1 ¼f0sg, Qn(cid:2)2 ¼U1, P1 ¼0 (because r¼0), Y1 ¼0, and
Algorithm 1b (if r>s), where r is the degree of the U2 ¼0 (because m¼s). Then,
modulating polynomial M and s is the number of bits
n(cid:2)3 n(cid:2)2
contained in each tuple Qi. The code C depends on the Y2 ¼Y ¼XQiþU1 ¼XQi:
components such as the parameters r;m;s;n, the weights
i¼0 i¼0
Wne0n;tWs 1w;.il.l.;pWrond(cid:2)u1,ceanddiftfhereencotdceodCe1s. TCh.uWs,edinfofewrenshtocwomtphoa-t From(7)ofAlgorithm1a,wehaveP2 ¼Y ¼Pin¼(cid:2)02Qi.Thus,
theresultingcodeCistheblock-paritycodepresentedin[4].
Algorithm 1 produces the codes in Fig. 1 by letting C1 be
trivialcodessuchasðs;s;1)andf0sgdefinedinRemark2.The 3.4 Cyclic Redundancy Code (CRC)
algorithm also produces other linear and nonlinear codes
Consideranh-bitCRCthatisqbitslongandisgeneratedbya
(Sections3.1,3.6,and3.8).Generally,codesofdC ¼4require polynomial M. Suppose that q and h can be written as q¼
thatn(cid:1)2randtheweightsW0;W1;...;Wn(cid:2)1inAlgorithm1 xþðn(cid:2)1Þs and h¼asþb, where n(cid:3)1, 0(cid:1)x(cid:1)s, a(cid:3)0,
bedistinct.CodesofdC ¼3requirethatn(cid:1)2rþ1andallow and 0<b(cid:1)s (see Appendix B, which can be found on the
someoftheweightstoberepeated.Codesofd ¼2alsoallow
C
Computer Society Digital Library at http://computer.org/
someoftheweightstoberepeated,butdonotrestrictonthe
tc/archives.htm). Then, it is shown in Remark B1 that the
value of n, i.e., the code lengths can be arbitrary. The
CRCchecktupleis
followingcodesarebrieflypresentedbecausetheirdetailed
discussionscanbefoundelsewhere[4],[5],[9],[14]. n(cid:2)a(cid:2)2 !
X
P ¼ QW þU Xr modM;
3.1 Binary Extended Perfect Code i i 1
i¼0
We now show that Algorithm 1 produces an extended
perfect code if the code C1 is an extended perfect code. whereWi ¼Xðn(cid:2)1(cid:2)iÞsmodM,i¼0;1;...;n(cid:2)a(cid:2)2.Further,
caSonuddpeph(os¼seeert[þh8,amCt h,Cat1hpeitsnerta6h]eð)2,cmio.(cid:2)ed.1,e;s2Cm¼(cid:2)h21ams(cid:2)(cid:2)nm1sa;¼n4dÞ2rdeþ1xmt¼e(cid:2)n14d¼.eLd2ehtp(cid:2)n1er¼bfiet2csrt. 2awhree(cid:2)1sd(cid:2)hios1wtinaicnntdRaMnemdiasWrktihB6¼e1p0t;rho1a;dtXutcsh;te.o.w.f;eðXiXgahsþt,in1pgÞroapvnoiddlyenadopmtrhiiamatlsitqiWv(cid:1)ei
Then, dC ¼4 by Theorem 1.2 and C is a ð2h(cid:2)1;2h(cid:2)1(cid:2)h;4Þ polynomial of degree h(cid:2)1.
Now, consider the code C that has the same length and
extended perfect code. Note that deleting a check bit from
thesameweightingpolynomialsastheaboveCRC.Letr¼
an extended perfect code will yield a perfect code, while
addinganoverallevenparitybittoaperfectcodewillyield h and m¼0. Then, P2 ¼0 by Remark 2.2 and P1 ¼P by
an extended perfect code. Algorithm 1a (if r(cid:1)s) or by Algorithm 1b (if r>s). Thus,
Algorithms 1a and 1b can be further generalized to thisparticularcodeC isidenticaltotheaboveCRC.So,any
include the extended perfect code of [15] as follows: Recall CRC can be generated by either Algorithm 1a or
thatP1,P2,andY1 arethecheckr-tuple,checkm-tuple,and Algorithm 1b, i.e., by Algorithm 1.
ðs(cid:2)mÞ-tuple, respectively, which are computed from Remark 3. To construct other codes (such as CXOR
Algorithms 1a or 1b. Let Eð:Þ be any function from the set checksum and nonbinary Hamming codes), we need to
ofðs(cid:2)mÞ-tuplestothesetofr-tuples.Now,definethenew modify(3)bydeletingQn(cid:2)2 fromthesummation,but(2)
check r-tuple and check m-tuple by remains unchanged. That is, (3) is replaced by
P1(cid:6) ¼P1þEðY1ÞandP2(cid:6) ¼P2þ evenparityof EðY1Þ: Xn(cid:2)3 !
Q þQ 2C : ð9Þ
Then,itcanbeshownthat,ifC1isanextendedperfectcode i¼0 i n(cid:2)1 1
and n¼2r, then the resulting code C whose check tuples
are P(cid:6) and P(cid:6) is also an extended perfect code. Further, Then, Algorithm 1a remains valid if we define Y ¼
when1 r¼1, 2this extended perfect code becomes the Pin¼(cid:2)03QiþU2Xm because the term Qn(cid:2)2 ¼U1XrþP1 is
extended perfect code that is obtained from the systematic absent from (9).
perfect code of [15].
3.5 CXOR Checksum
3.2 Weighted Sum Code (WSC) Suppose now that we allow some of the polynomials
Consider the code C for the special case s¼r¼m. By W0;W1;...;Wn(cid:2)1 in (2) to be repeated and we use
Remark2.1,we haveC1 ¼f0sg,U1 ¼0,U2 ¼0,Y1 ¼0,and Algorithm 1a along with variation (9). Let r¼s¼m,
Y2 ¼Y ¼Pni¼(cid:2)03QiþP1. From (6) and (7) of Algorithm 1a, M ¼Xsþ1, and Wi ¼XimodM. It can be shown that
we have Wiþs ¼Wi for all i(cid:3)1, i.e., some of the weighting
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6 IEEETRANSACTIONSONCOMPUTERS, VOL.54, NO.1, JANUARY2005
polynomials may repeat. Then, C1 ¼f0sg (because m¼s), ð4;096;4;080;4Þ nonlinear code that is twice as
U1 ¼0 (because r¼s), and U2 ¼Y1 ¼0 (because m¼s). long as the linear code in 1.
From (6) and (7), we have P1 ¼Pni¼(cid:2)03QiXimodðXsþ1Þ
and P2 ¼Y2 ¼Y ¼Pni¼(cid:2)03Qi (see Remark 2.1 and 4 FAST IMPLEMENTATION OF ERROR-DETECTION
Remark 3). Thus, the resulting code C is the CXOR CODES
checksum presented in [4].
Recall from Algorithm 1 that r is the degree of the
3.6 Nonbinary Perfect Code modulating polynomial M and s is the number of bits
Suppose that Algorithm 1a along with variation (9) is contained in each tuple Qi. Algorithm 1 produces a large
applied with r¼s¼m and n¼2mþ1. Let M be a family of error-detection codes because its systematic
primitive polynomial of degree m and let versions (either Algorithm 1a when r(cid:1)s or Algorithm 1b
whenr>s)generateallthecodespresentedinSection3.So
W0;W1;...;Wn(cid:2)3 far, the discussion is abstract and general to facilitate the
be distinct and nonzero polynomials. Then, C1 ¼f0sg, developmentofmathematicalpropertiesofouralgorithms.
P1 ¼Pi2¼m0(cid:2)2QiWimodM, and P2 ¼P2i¼m0(cid:2)2Qi. It then can Ianlgothriitshmsesc,tiio.en.,,wweenfoowcudsisocnustshehopwrascotimcaelcaosdpeesctgeonferthateesde
be shown that P1 and P2 are two check tuples of the
nonbinary Hamming perfect code over GFð2mÞ (see [8, by these algorithms can be efficiently implemented in
software. Then, we compare the complexity of our
Chapter 6]), i.e., the tuples ðQ0;Q1;...;Q2m(cid:2)2;P1;P2Þ form
algorithms with that of the CRC algorithm (the strongest
the codewords of the Hamming perfect code over GFð2mÞ.
code in Fig. 1). In theory, the fundamental unit for digital
3.7 One’s-Complement Checksum and Fletcher data is bit. In practice, however, communication protocols
Checksum andcomputersoftenprocessdataasblocksofbitsortuples
The above codes are constructed using polynomial arith- (e.g., bytes or words) and not as individual bits at a time.
meticbecauseeachtupleisconsideredasapolynomialover For example, on familiar 32-bit computers, the modulo-2
the binary field f0;1g. An alternative is to consider each addition of two 32-bit numbers can be accomplished by a
single XOR operation (using C programming language).
tuple as an integer and to use the rules of (one’s-
Thus, efficient error-detection codes should also be pro-
complement) integer arithmetic to manipulate the code
cessedintermsoftuplesatatime,i.e.,eachns-bitcodeword
construction. If we apply the integer arithmetic to the
is expressed in terms of n tuples, s bits each.
construction of the block-parity code and to the nonbinary
In parallel to Algorithm 1a and Algorithm 1b, now we
perfect code, we will get the one’s-complement checksum
develop two fast algorithms: Algorithm 2a for r(cid:1)s and
and Fletcher checksum, respectively. However, these
Algorithm2bforr>s.AlthoughAlgorithms1aand1bcan
integer-based codes are often weaker than their binary
produce CRCs and many other codes (see Section 3), the
polynomial counterparts (see Fig. 1). See [4], [5], [9] for
two fast algorithms produce only non-CRC codes that are
definitionsandperformancecomparisonsoferror-detection
shown later in Section 4.1 to be faster than CRCs by the
codes,includingtheone’s-complementandFletchercheck-
factor OðsÞ.
sums.Thus,theinteger-arithmeticversionofAlgorithm1a,
Now, suppose that information tuples
along with variation (9), also produces the one’s-comple-
ment and Fletcher checksums. We will not discuss these ðQ ;Q ;...;Q ;U ;U Þ
0 1 n(cid:2)3 1 2
checksums and integer-based codes any further because
they are often weaker than their polynomial counterparts are given. Let r, s, and m be such that r(cid:1)s and n(cid:1)2r.
and their analyses can be found elsewhere (e.g., [5], [14]). Assume that each Qi is s bits, U1 is s(cid:2)r bits, and U2 is
s(cid:2)m bits. From the following algorithm, we can compute
3.8 Other Error-Detection Codes the two check tuples P1 and P2 that are appended to the
Recall from Algorithms 1a and 1b that the ðns;ns(cid:2)r(cid:2) information tuples such that the resulting code C has
m;dCÞcodeC isconstructedfromanðs;s(cid:2)m;d1ÞcodeC1. minimum distance dC ¼4.
Thus, by varying C1, different codes C are produced. Algorithm 2a. Let r(cid:1)s and n(cid:1)2r. Let M be a primitive
Further,CisnonlinearifC1isnonlinear.Thusfar,thecodes polynomialofdegreerandletF betheencoderofanðs;s(cid:2)
C are constructed from the codes C1 that are either m;d1Þ code with d1 (cid:3)4. Then, the resulting code C is an
extended perfect codes or trivial codes f0sg and ðs;s;1Þ. ðns;ns(cid:2)r(cid:2)m;4Þ code and each of its codewords is
Now, we construct the codes C from the codes C1 that are ðQ0;Q1;...;Qn(cid:2)3;U1;U2;P1;P2Þ. The two check tuples are
neither perfect nor trivial. In both instances below, we computed by
assume that s¼rþm¼16, n¼2r with r¼7 or 8, and
d1 ¼6, so that dC ¼4 by Theorem 1.2. P1 ¼ðZþU1XrÞmodM and P2 ¼Y2þFðY1Þ;
1. SupposethatC1 istheextendedð16;7;6ÞlinearBCH whereZ ¼Pni¼(cid:2)03QiXn(cid:2)2(cid:2)imodðMXs(cid:2)rÞ.ThetuplesY1 and
code(see[8],Chapter3)andr¼7.Then,ns¼2rs¼ Y2 are defined as in Algorithm 1a, i.e., they satisfy
2;048 and the resulting code C is a ð2;048;2;032;4Þ ðY1;Y2Þ¼Y1XmþY2 ¼Y ¼Pni¼(cid:2)03QiþU1XrþP1þU2Xm.
linear code. Proof. Define Wi ¼Xn(cid:2)2(cid:2)imodM, i¼0;1;...;n(cid:2)3. Then,
2. Suppose that C1 is the extended ð16;8;6Þ nonlinear Wi 6¼0;1andW0;W1;...;Wn(cid:2)3 aredistinctbecauseM is
Nordstrom-Robinson code (see [8, p. 73]) and aprimitivepolynomialofdegreerandn(cid:1)2r.LetC1 be
r¼8. Then, ns¼2rs¼4;096, and C is a the ðs;s(cid:2)m;d1Þ code with the encoder F. Now, using
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NGUYEN: ERROR-DETECTIONCODES:ALGORITHMSANDFASTIMPLEMENTATION 7
these Wi and C1 in Algorithm 1a, we can construct the
code C whose two check tuples are given by
!
n(cid:2)3
X
P ¼ QW þU Xr modM and P ¼Y þFðY Þ:
1 i i 1 2 2 1
i¼0
Because d1 (cid:3)4, dC ¼4 by Theorem 1.2. Next, the new
form of P1 is derived as follows: First, note that
Xs(cid:2)rW ¼ðXs(cid:2)rXn(cid:2)2(cid:2)iÞmodðMXs(cid:2)rÞ: ð10Þ
i
Multiplying P1 by Xs(cid:2)r, we have
!
n(cid:2)3
X
P Xs(cid:2)r ¼ QW þU Xr Xs(cid:2)rmodðMXs(cid:2)rÞ: ð11Þ
1 i i 1
i¼0
From (10), (11), the definition of Z, and some
modular algebra manipulation, it can be shown that
P1Xs(cid:2)r ¼ðXs(cid:2)rZþU1XrXs(cid:2)rÞmodðMXs(cid:2)rÞ. Thus, Fig. 3. Pseudocode for Algorithm 2a. Here, Qi, U1, and U2 are input
informationtuples,P istheoutputchecktuple.
P Xs(cid:2)r ¼ððZþU XrÞmodMÞXs(cid:2)r: ð12Þ
1 1
From (12), we have P1 ¼ðZþU1XrÞmodM. tu Remark 5. Algorithm 2b, which will be discussed
shortly, requires operations on new tuples
Remark 4. Let I and J be two binary polynomials of
Q(cid:6);Q(cid:6);...;Q(cid:6) that are defined from the original
degrees i and j, respectively. In general, it is rather 0 1 n(cid:2)3
tedious to compute ImodJ. However, when i(cid:1)j, the tuples Q0;Q1;...;Qn(cid:2)a(cid:2)3 as follows: First, let
U¼f0;1;...;n(cid:2)4;n(cid:2)3g, then we partition the set U
computation becomes easy and is accomplished in
constant time because into four sets P, Q, X, and Y:
(cid:3)I if i<j P¼fi:0(cid:1)i(cid:1)n(cid:2)3(cid:2)a;i¼n(cid:2)2(cid:2)js
ImodJ ¼
IþJ if i¼j: forsomej;1(cid:1)j(cid:1)ag;
Q¼fi:0(cid:1)i(cid:1)n(cid:2)3(cid:2)a;i6¼n(cid:2)2(cid:2)js
This simple fact is used to efficiently compute Z ¼
Pin¼(cid:2)03QiXn(cid:2)2(cid:2)imodðMXs(cid:2)rÞinAlgorithm2a,asfollows: forallj;1(cid:1)j(cid:1)ag;
Using Horner’s rule, we have X¼fi:n(cid:2)3(cid:2)a<i(cid:1)n(cid:2)3;i¼n(cid:2)2(cid:2)js
forsomej;1(cid:1)j(cid:1)ag;
Z ¼ð...ððQ XÞmodNþQ ÞXmodNþ...þQ ÞmodN;
0 1 n(cid:2)3
Y¼fi:n(cid:2)3(cid:2)a<i(cid:1)n(cid:2)3;i6¼n(cid:2)2(cid:2)js
where N ¼MXs(cid:2)r. Then, Z can be recursively
forallj;1(cid:1)j(cid:1)ag:
computed from the polynomials Ti defined by T0 ¼
Q0 and Ti ¼ðTi(cid:2)1XÞmodNþQi, i¼1;...;n(cid:2)3. Because jPjþjXj(cid:1)a and a¼jXjþjYj, we have
Because s¼degN (cid:3)degðTi(cid:2)1XÞ, each Ti is computed jPj(cid:1)jYj. Let p¼jPj, then Y has at least p elements.
in constant time, i.e., with Oð1Þ complexity. Finally, we So, let Y(cid:6) be the set of p smallest elements of Y, i.e.,
have Z ¼ðTn(cid:2)3XÞmodN. Thus, Z has computational Y(cid:6) ¼ff1;f2;...;fpg. Similarly, we can write
complexityOðnÞ.Horner’sruleisalsousedtoefficiently P¼fe1;e2;...;epg.Finally,wecandefinethenewtuples
encode the WSC [4], [9]. Q(cid:6)0;Q(cid:6)1;...;Q(cid:6)n(cid:2)3 from Q0;Q1;...;Qn(cid:2)a(cid:2)3 as follows:
Fig. 3 shows a simple software implementation of 8
0 if i2P[X
Algorithm 2a. The input data are ðQ0;Q1;...;Qn(cid:2)3;U1;U2Þ. Q(cid:6) ¼<Q if i2Q
TheoutputisthechecktupleP ¼ðP1;P2Þ.The“for”loopis i :0i if p<i(cid:1)n(cid:2)3;
usedtocomputebothY andT.ComputationofY requires
coonmlypuotnede vXiOaRReompaerrkat4iobne,cawuhseiledeTgNca(cid:3)ndebgeðTeXffÞi.ciTehnetlny, and Q(cid:6)fi ¼Qei if 1(cid:1)i(cid:1)p.
Remark 6. Now, assume that s(cid:3)aþ1 and n(cid:3)2þas,
the final value of T is used to compute Z, i.e.,
we will show that these conditions will simplify
Z ¼ðTXÞmodN.
the definition of Q(cid:6) (given in Remark 5). It can be
Now, we consider a fast version of Algorithm 1b for i
constructing the code C. In this case, r>s, i.e., r¼asþb, shown from these conditions that 0(cid:1)n(cid:2)2(cid:2)js(cid:1)
where a(cid:3)1 and 0<b(cid:1)s. Assume that information n(cid:2)3(cid:2)a for all 1(cid:1) j(cid:1)a. Then, from Remark 5,
tuples ðQ0;Q1;...;Qn(cid:2)a(cid:2)3;U1;U2Þ are given, where each we have p¼jPj¼a, jXj¼0, jYj¼a, and Y(cid:6) ¼Y.
Qi is s bits, U1 is s(cid:2)b bits, and U2 is s(cid:2)m bits. We wish We also have Y(cid:6) ¼fn(cid:2)2(cid:2)j;j¼a;a(cid:2)1;...;1g and
to append two checks tuples P1 and P2 to the information P¼fn(cid:2)2(cid:2)js;j¼a;a(cid:2)1;...;1g. Note that fi ¼n(cid:2)
tuples so that ðQ0;Q1;...;Qn(cid:2)a(cid:2)3;U1;U2;P1;P2Þ is a code- 2(cid:2)ðaþ1(cid:2)iÞ and ei ¼n(cid:2)2(cid:2)ðaþ1(cid:2)iÞs, 1(cid:1)i(cid:1)a.
word of the code C. Before stating the algorithm, we Thus, Q(cid:6)fi ¼Qei iff Q(cid:6)n(cid:2)2(cid:2)ðaþ1(cid:2)iÞ ¼Qn(cid:2)2(cid:2)ðaþ1(cid:2)iÞs iff
need some preliminary results. Q(cid:6)n(cid:2)2(cid:2)js ¼Qn(cid:2)2(cid:2)js. Finally, from Remark 5, we have
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8 IEEETRANSACTIONSONCOMPUTERS, VOL.54, NO.1, JANUARY2005
Q(cid:6) ¼Q if i6¼n(cid:2)2(cid:2)js;1(cid:1)j(cid:1)a;0(cid:1)i(cid:1)n(cid:2)3(cid:2)a;
i i
Q(cid:6) ¼0;1(cid:1)j(cid:1)a; and
n(cid:2)2(cid:2)js
Q(cid:6) ¼Q ;1(cid:1)j(cid:1)a:
n(cid:2)2(cid:2)j n(cid:2)2(cid:2)js
Basically, Q(cid:6);Q(cid:6);...;Q(cid:6) are obtained by moving
0 1 n(cid:2)3
some a tuples of Q0;Q1;...;Qn(cid:2)a(cid:2)3 to the right and
then by filling the a removed tuples by zeros. Now,
define Z ¼Pn(cid:2)3Q(cid:6)Xn(cid:2)2(cid:2)imodM, which is a key
i¼0 i
quantity in the following algorithm. This simplified
definition of Q(cid:6) makes it possible to calculate Z directly
i
from Qi (i.e., without usingQ(cid:6)i). That is,we first modify
Qi using the following pseudocode:
forð1(cid:1)j(cid:1)aÞQ ¼Q ;
n(cid:2)2(cid:2)j n(cid:2)2(cid:2)js
forð1(cid:1)j(cid:1)aÞQ ¼0;
n(cid:2)2(cid:2)js
thenwecancomputeZ directlyfromthemodifiedQi as
Z ¼Pni¼(cid:2)03QiXn(cid:2)2(cid:2)imodM.
Now, we have the following algorithm that is proven in
AppendixA(whichcanbefoundontheComputerSociety Fig. 4. Pseudocode for Algorithm 2b. Here, Qi, U1, and U2 are input
Digital Library at http://computer.org/tc/archives.htm). informationtuples,P istheoutputchecktuple.
Algorithm 2b. Suppose that r>s and n(cid:1)2r. Let M be a
Inthisexample,a¼1,s¼8,andnisthetotalnumber
primitivepolynomialofdegreerandletF betheencoderof
of bytes in a codeword of the code C. If we assume
an ðs;s(cid:2)m;d1Þ code with d1 (cid:3)4. Then, the two check furtherthatn(cid:3)10,thenthehypothesesofRemark6are
tuples of the code C are computed by satisfied,i.e.,s(cid:3)aþ1andn(cid:3)2þas.Thus,byRemark
P ¼ðZþU XrÞmodM and P ¼Y þFðY Þ; 6,wecanmodifytheQibyfirstsettingQn(cid:2)3 ¼Qn(cid:2)10and
1 1 2 2 1 then setting Qn(cid:2)10 ¼0. Then, the modified information
where Z ¼Pn(cid:2)3Q(cid:6)Xn(cid:2)2(cid:2)imodM and Q(cid:6) are defined in tuples are
i¼0 i i
Remark 5 (or in Remark 6 if applicable). The tuples Y1 and
ðQ ;Q ;...;Q ;Q ;Q ;...;Q ;Q ;ðU ;U ÞÞ:
Y2 are defined as in Algorithm 1b, i.e., they satisfy 0 1 n(cid:2)11 n(cid:2)10 n(cid:2)9 n(cid:2)4 n(cid:2)3 1 2
Then, as in Algorithm 2a, we can efficiently compute
ðY ;Y Þ¼Y XmþY ¼Y
1 n2(cid:2)a(cid:2)3 1! 2 a ! the quantity Z ¼Pni¼(cid:2)03QiXn(cid:2)2(cid:2)imodM, which is
¼ X Q þ(cid:1)U XbþP (cid:2)þ XP þU Xm; shown in Fig. 4.
i 1 10 1i 2
i¼0 i¼1 Remark 7.
where P10 is a b-tuple, and P11;...;P1a are s-tuples that 1. Givenrands,eitherAlgorithm2aorAlgorithm2b
satisfy P1 ¼ðP10;P11;...;P1aÞ. Further, C is an ðns;ns(cid:2) can be used to construct the code C that is ns
r(cid:2)m;4Þ code. bitslong,where1(cid:1)n(cid:1)2r.Thevaluesofrands
Fig. 4 shows a software implementation of
canbeassmallas0and1,respectively.However,
Algorithm 2b under the assumption that s(cid:3)aþ1 and the resulting code C can be trivial, e.g., if r¼0,
n(cid:3)2þas as required in Remark 6. The input data are then n¼1 and C ¼C1. If r¼0, s¼1, and
Q0;Q1;...;Qn(cid:2)a(cid:2)3;U1;U2. The output is the check tuple C1 ¼f0g, then C ¼C1 ¼f0g. If s¼1, C1 ¼f0g,
P ¼ðP1;P2Þ. Note that, as in Algorithm 2a (see r¼1, and n¼2r ¼2, then C ¼fð0;0Þg. How-
Remark 4), the tuple Z in Algorithm 2b can also be ever, when s¼1, C1 ¼f0g, r(cid:3)2, and n(cid:3)4, the
computed in time OðnÞ. resulting code C can be nontrivial and each
Example3.Here,weconstructthecodeCforthecaser>s, codewordofC nowhasns¼nbits.Inparticular,
from Algorithm 2b, it can be shown that the two
with s¼8 and r¼12. Let m¼4, then the total number
check tuples of an n-bit codeword are
of check bits is h¼rþm¼16. Because r>s, we can
write r¼asþb with a¼1 and b¼4. Let n(cid:2)a(cid:2)3 a
X X
ðQ0;Q1;...;Qn(cid:2)4;U1;U2Þ be information tuples, where P1 ¼Z and P2 ¼ Qiþ P1i;
each Qi is an 8-tuple, U1 and U2 are 4-tuples, which can i¼0 i¼0
be combined into a single 8-tuple ðU1;U2Þ. We wish to i.e.,P2istheevenparitybitcomputedfromthefirst
append a check 16-tuple ðP1;P2Þ to the information n(cid:2)1 bits of the codeword of C. For example, if
tuples so that ðQ0;Q1;...;Qn(cid:2)4;ðU1;U2Þ;P1;P2Þ forms a r¼2andn¼2r ¼4,thenC istheð4;1;4Þrepeti-
codeword of the code C, which is an ð8n;8n(cid:2)16;4Þ
tion code. This ð4;1;4Þ code is also constructed
code. Here, we let F be the encoder of the ð8;4;4)
extendedHammingcode.TheresultingcodeC canhave from Algorithm 2a with r¼1, n¼2r ¼2, s¼2,
length up to 2h(cid:2)1 ¼215 bits (see Section 3.1). andC1 ¼fð0;0Þg.
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