New Error Correcting Codes for Informed Receivers

0.31 MB·

by  Lakshmi Natarajan

1
New Error Correcting Codes for Informed Receivers
Lakshmi Natarajan, Yi Hong, and Emanuele Viterbo
Abstract—We construct error correcting codes for jointly In this paper, we construct several families of linear block
transmitting a finite set of independent messages to an informed codes for informed receivers by using algebraic error cor-
receiver which has prior knowledge of the values of some subset
recting codes. The constructed coding schemes are of modest
ofthemessages assideinformation.The transmitterisoblivious
blocklengths,mostofthepresentedschemesprovideoptimum
tothemessagesubsetalreadyknowntothereceiverandperforms
encodinginsuchawaythatanypossiblesideinformationcanbe or near-optimum error correction capability and guarantee
usedefficientlyatthedecoder.Weconstructandidentifyseveral largerminimumdistancesthanknowncodingschemesofsim-
families of algebraic error correcting codes for this problem ilar rate and block length available from [1]. After illustrating
using cyclic and maximum distance separable (MDS) codes.
6 theproblemwithanexampleusinganewoptimalbinarycode
The proposed codes are of short block length, many of them
1 forinformedreceivers(SectionIII),wecharacterizethefamily
provide optimum or near-optimum error correction capabilities
0
and guarantee larger minimum distances than known codes of of maximal distance separable (MDS) codes for this channel
2
similarparametersforinformedreceivers.Theconstructedcodes (Section IV). We then construct and identify several families

r are also useful as error correcting codes for index coding when of binary codes for informed receivers using cyclic codes,
p
the transmitter does not know the side information available at
A quadratic and cubic residue codes, and concatenated coding
the receivers.
(Section V).

7 Index Terms—Cyclic codes, index coding, informed receivers, A related problem is that of index coding [9] where L mes-
maximum distance separable codes, side information.
sagesaretobebroadcasttoasetofreceiversandeachreceiver
]
T demands some subset of the source messages while having
I I. INTRODUCTION prior knowledge of a different subset as side information. The
.
s error correcting codes for this problem available in [10], [11]
c We consider the channel coding problem where the trans-
assumethatthedemandsandsideinformationofthereceivers
[ mitter jointly encodes a set of L independent messages while
are known to the transmitter. On the other hand, the codes of
the receiver has prior knowledge of the values of some subset
2 this paper are suitable when no such knowledge is available
v of the L messages. The transmitter is ignorant of the subset
at the encoder.
8 of source messages already known at the receiver, and hence,
The codes C constructed in this paper decompose as a
6 is required to encode the messages in such a way that every
4 direct sum of L subcodes C1,...,CL such that each of the
possible side information at the receiver can be exploited effi-
6 2L−1 subcodes of C formed as the sum of some subset of
ciently.Following[1],werefertothiscommunicationproblem
0 {C ,...,C } is a good error correcting code for its rate and
1. ascodingforinformedreceiver.Anequivalentcommunication blo1cklengthL. Constructions of pairs C ⊃C(cid:48), or even chains
scenario is the broadcast of L messages to multiple receivers
0 C ⊃ C(cid:48) ⊃ C(cid:48)(cid:48) ⊃ ···, of linear codes have been previously
6 where each receiver has side information of a different subset investigated in the literature; see, for example, [6], [12], [13].
1 of source messages. Applications of this problem include the
However, these nested codes are not useful when the receiver
: broadcast phase of decode-and-forward protocol in multi-way
v side information is an arbitrary subset of source messages.
Xi relay networks [2], [3], and retransmissions in a broadcast Notation: Matrices and row vectors are denoted by bold
channel where each receiver has successfully decoded some
upperandlowercaseletters,respectively.TheminimumHam-
ar subset of the messages from previous transmissions [4], [5]. ming distance of a code C is denoted by d(C). The symbol
It is known that linear coding schemes for informed receivers
d∗(n,k) denotes the maximum of the minimum Hamming
involve the design of L codes C ,...,C which are linearly
1 L distances over all [n,k] binary linear codes. Unless otherwise
independent(asvectorspaces)suchthatthesumofanysubset
statedallthevaluesof,andboundson,d∗(n,k)arereferenced
of the L codes is a good error correcting code [1], [2]. To the
from the table of best known linear codes available in [14].
best of our knowledge, only a few explicit constructions of
codesforinformedreceiversbasedonconvolutionalcodes[6], II. REVIEWOFLINEARANDCYCLICCODES
[7], LDPC codes [8], and errors-and-erasures decoding of
Let q be a prime power and F be the finite field of size
linear codes [1] are available. q
q. In order to design coding schemes for receivers with side
information, we will consider finite collections of length n
The authors are with the Department of Electrical and Computer
Systems Engineering, Monash University, VIC 3800, Australia (e-mail: linear codes over Fq. A collection {C1,...,CL} of linear
{lakshmi.natarajan,yi.hong,emanuele.viterbo}@monash.edu).Thisworkwas codes is linearly independent if the only choice of ccc ∈ C ,
(cid:96) (cid:96)
supportedbytheAustralianResearchCouncilunderGrantDiscoveryProject (cid:96) = 1,...,L, satisfying (cid:80)L ccc =000 is ccc = ··· =ccc =000.
No.DP160101077. (cid:96)=1 (cid:96) 1 L
©2016 IEEE. Personal use of this material is permitted. Permission from If C1,...,CL are linearly independent, the sum code C =
IEEE must be obtained for all other uses, in any current or future media, (cid:80)L C is their direct sum as a vector space.
includingreprinting/republishingthismaterialforadvertisingorpromotional (cid:96)=1 (cid:96)
We now introduce the notation and briefly review some
purposes,creatingnewcollectiveworks,forresaleorredistributiontoservers
orlists,orreuseofanycopyrightedcomponentofthisworkinotherworks. of the relevant properties of cyclic codes based on [15],
2
[16]. We consider cyclic codes of length n over F with todenotethelinearcodewithgeneratormatrixGGG ,weobserve
q (cid:96)
gcd(n,q)=1. Label the coordinates of ccc∈Fn with the that C ,...,C are linearly independent and C is their direct
q 1 L
elements of Z ={0,1,...,n−1} and associate the vector sum.
n
ccc = (c ,...,c ) with the polynomial c(x) = c +c x+ ForS (cid:40){1,...,L},considerthereceiverRx thathasprior
0 n−1 0 1 S
···+c xn−1.WiththiscorrespondenceacycliccodeC isan informationofthevaluesofwww ,(cid:96)∈S.Notethatthisincludes
n−1 (cid:96)
idealintheringR =F [x]/(xn−1).Weuseg(x)todenote the case S =∅, i.e., no side information. On observing the
n q
the generator polynomial of C and h(x)=(xn−1)/g(x) to channel output yyy = ccc+zzz, where zzz is the error vector, Rx
S
denote its check polynomial. removes the contributions of www , (cid:96)∈S, from yyy to arrive at
(cid:96)
The q-cyclotomic coset modulo n of i∈Zn is the set yyy =yyy−(cid:88)www GGG =(cid:88)www GGG +zzz,
C ={i,qi,q2i,...},wherethearithmeticisperformedmod- S (cid:96) (cid:96) (cid:96) (cid:96)
i
ulo n. Let α be a primitive nth root of unity in some (cid:96)∈S (cid:96)∈S¯
extension field F ⊃F . Then, for some T ⊂Z , we have where S¯ is the complement of S in {1,...,L}. The unknown
qm q n
h(x)=(cid:81) (x−αi) and g(x)=(cid:81) (x−αi), where T is messages www , (cid:96)∈S¯, are then estimated by decoding yyy to
i∈T i∈T (cid:96) S
tzheeroceosmapnldemzeernotesofofTCin, rZensp.eTchtievesleyt,saTndaenadchTisarae uthneionnono-f CS¯ =(cid:88)C(cid:96) =(cid:110)(cid:88)www(cid:96)GGG(cid:96)(cid:12)(cid:12)(cid:12)www(cid:96) ∈Fkq,(cid:96)∈S¯(cid:111).
q-cyclotomic cosets modulo n. The dimension of C is |T|. (cid:96)∈S¯ (cid:96)∈S¯
When q =2, the subcode of C consisting of all even weight Note that CS¯ is a subcode of C of dimension k|S¯|. In order
codewords of C is a cyclic code with non-zeroes T \{0}. A to maximize the error correction capability at Rx we require
S
cyclic code is irreducible if it contains no non-trivial cyclic that CS¯ be a good linear error correcting code.
subcodes. A cyclic code with non-zeroes T is irreducible if We are interested in the scenario where the transmitter is
and only if T is a cyclotomic coset by itself. oblivious to the side information S available at the receiver,
IfT1,...,TL arethenon-zeroesofcycliccodesC1,...,CL andthus,werequirethateachofthe2L−1subcodesCS¯ ofthe
then ∪L T is the set of non-zeroes of C = (cid:80)L C . With code C, one corresponding to each possible side information
(cid:96)=1 (cid:96) (cid:96)=1 (cid:96)
thenotationasabove,westatethefollowingfactwhoseproof configurationS (cid:40){1,...,L},beagooderrorcorrectingcode,
is straightforward. i.e., with a large minimum Hamming distance d(CS¯).
Lemma 1. The collection {C ,...,C } is linearly indepen- Definition 1. Anerrorcorrectingcodeforinformedreceivers
1 L
dent if and only if T ,...,T are non-intersecting. (ECCIR) encoding L messages is a linearly independent
1 L
collection {C ,...,C } of L linear codes.
Let a be any integer with gcd(a,n)=1. The function 1 L
i→ai mod n, is a permutation on the set of coordinates Zn Thecodedesignobjectiveistoconstruct{C1,...,CL}such
since a has a multiplicative inverse in Zn. The polynomial that all the minimum Hamming distances d(CS¯), for every
µ (c(x)) obtained by applying this permutation on the coor- S (cid:40){1,...,L}, are as large as possible.
a
dinates of c(x) is
Example 1. A new optimum binary ECCIR of length n=31
c(xa)=c +c xa+···c xa(n−1) mod (xn−1). for L=3 messages of size k =10 each: Consider cyclic
0 1 n−1
codes of length n=31 over the binary alphabet q =2. The
When applied to a cyclic code (ideal) C ⊂Rn, µa(C) is the cyclotomic cosets are
set of all polynomials µ (c(x)) with c(x) ∈ C, and further,
a
µ (C) is itself a cyclic code. C1 ={1,2,4,8,16}, C3 ={3,6,12,24,17}
a
C ={5,10,20,9,18}, C ={7,14,28,25,19}
Lemma 2. The set of non-zeroes of C is T if and only if the 5 7
set of non-zeroes of µ (C) is a−1T. C11 ={11,22,13,26,21}, C15 ={15,30,29,27,23},
a
Proof: An element j ∈Z is a non-zero of a cyclic code and C0 = {0}. The [31,30,2] single-parity check code C is
n
thecycliccodewithnon-zeroesT ={1,...,30}.Considerthe
if and only if there exists a codeword polynomial for which
αj is not a root. If j ∈T, then there exists c(x) ∈ C such codes C(cid:96), (cid:96) = 1,2,3, with non-zeroes T1 = C1 ∪C3, T2 =
that c(αj) (cid:54)= 0. The evaluation of c(cid:48)(x) = c(xa) ∈ µ (C) at C5∪C15 and T3 =C7∪C11, respectively. Since T1∪T2∪T3
x = αa−1j yields c(cid:48)(αa−1j) = c(αj) (cid:54)= 0 showing thaat a−1j is a partition of T, C is a direct sum of C(cid:96), (cid:96)=1,2,3 (from
is a non-zero of µ (C). The proof of converse is similar. Lemma 1).
a
Equivalence of codes: Since n=31 is prime,
gcd(a,n)=31 for any non-zero a∈Z . Observe that
III. ERRORCORRECTIONFORINFORMEDRECEIVERS T =25T =9T , and 25−1 =5 and 9−1 =317 in Z . Using
1 2 3 31
Consider a vector www of length kL over Fq composed Lemma 2, we deduce that C1 =µ5(C2)=µ7(C3). It follows
of L independent message vectors www1,...,wwwL ∈Fkq, i.e., that C1,C2,C3 are equivalent up to coordinate permutations,
www =(www1,...,wwwL). The message www is encoded by a linear and in particular, they have the same minimum distance.
code C (not necessarily cyclic) of length n using a generator Similarly, since T ∪T = 9(T ∪T ) = 25(T ∪T ), the
1 2 1 3 2 3
matrix GGG∈FkqL×n of full-rank kL. Let GGG1,...,GGGL ∈Fkq×n three codes C1+C2, C1+C3 and C2+C3 are equivalent.
bethesubmatrice(cid:0)so(cid:124)fGGGcorre(cid:124)s(cid:1)p(cid:124)ondingtowww1,...,wwwL,respec- Minimum distance of C2+C3: The code C2+C3 (zeroes
tively, i.e., GGG = GGG ··· GGG . The message www is encoded C ∪ C ∪ C ) is the even weight subcode of the double-
1 L 0 1 3
into the length n codewordccc=wwwGGG=(cid:80)L www GGG . Using C error correcting BCH code (zeroes C ∪C ) with parameters
(cid:96)=1 (cid:96) (cid:96) (cid:96) 1 3
3
[31,21,5]. Hence, d(C +C )≥6. Since d∗(31,20)=6, we Theorem 1 ([15, p. 321]). Every square submatrix of GGG is
2 3
conclude that C +C is a [31,20,6] code, where d∗(n,k) nonsingularifandonlyifAAA=[III|GGG]generatesanMDScode.
2 3
is the maximum of the minimum Hamming distances over all
It follows that we can construct length n MDSIR codes
[n,k] binary codes.
for L symbols by puncturing the information coordinates of
Minimum distance of C : The code C is equivalent to
1 1 any [n+L,L] systematic MDS code, such as the systematic
µ (C ) (non-zeroes C ∪ C ). Note that −1=30 and
−1 1 −1 −3 versions of extended Reed-Solomon (RS) and generalized
−3=28 in Z . The dual of µ (C ) has zeroes at C ∪C ,
31 −1 1 1 3 RS codes. For example, a length n MDSIR code for L
and hence, (µ (C ))⊥ is the double-error correcting primi-
−1 1 messages and k =1 exists over all F with q >n+L since
tive BCH code with parameters [31,21,5]. Using the Carlitz- q
an[n+L,L,n+1]generalizedRScodeexistsoversuchF .
Uchiyama bound [15, p. 280], we know that the minimum q
distance of the dual of the [31,21,5] primitive BCH code is
even and satisfies the lower bound d(µ−1(C1)) ≥ 2(5−1) − Comparison with the ECCIRs of [1]
25/2 = 10.34.., i.e., d(µ (C )) ≥ 12. Using the fact that
−1 1 A construction of ECCIRs similar to that of this section
d∗(31,10) = 12, we conclude that µ (C ) and C are
−1 1 1 was proposed in [1] using an approach based on errors-and-
[31,10,12] codes.
erasures decoding of linear codes. For any q,k, [1] shows
In summary, C , C and C are all [31,10,12] codes, the
1 2 3 that if AAA=[III|GGG] is the generator of an [n+kL,kL,d]
sumofanytwoofthesethreecodesis[31,20,6],andthesum
code (not necessarily MDS) over F , then the L submatrices
of all three is the [31,30,2] code. All these codes have the GGG ,...,GGG of GGG=(GGG(cid:124),...,GGG(cid:124))(cid:124)qgenerate an ECCIR with
oApttitmheumrecmeiivneirm, uthme mdiisntaimncuemfodristthaneicresledn(gCthS¯)ancdorrdeismpeonnsdiionng. dM(1DCSS¯)c≥odmeL,ai.xe{.,dif−dk=|S¯n|1,+0}1.,WtheeLrecmonasrtkrutchtaiotnifoAAAf [g1e]nyeirealtdessaann
to the side information configuration S, with |S|=0,1,2, are
MDSIR code. Compared to [1], our approach illuminates the
2,6,12, respectively.
direct and strong relation between MDS and MDSIR codes
(Lemma 3 and Theorem 1).
IV. MAXIMUMDISTANCESEPARABLECODES As illustrated by the following example, the binary non-
InthissectionweconstructECCIRssuchthatallthe2L−1 MDSIR codes constructed in this paper (Example 1 and
codes CS¯, S (cid:40) {1,...,L}, meet their respective Singleton Section V) can guarantee larger minimum distances than the
bounds d(CS¯)≤n−k|S¯|+1. binary codes constructed using the technique of [1].
Definition 2. A collection of codes {C ,...,C } is maxi- Example 2. To generate an ECCIR with the same parameters
1 L
mum distance separable for informed receivers (MDSIR) if (k =10,L=3,n=31,q =2)asthenewcodeofExample1,
d(CS¯)=n−k|S¯|+1 for every S (cid:40){1,...,L}. the approach of [1] starts with the best known binary code of
length n+kL=61 and dimension kL=30, which has min-
WeconstructMDSIRcodes{C ,...,C }fork =1andar-
1 L imum distance d=12 [14]. For |S| = 0,1,2, this technique
banityrakryaLn,di.Le.,wwhitehrekeLach=mLes,saangeMwwwD(cid:96)SiIsRacsocdaela{rCby(cid:48),.it.s.e,lfC.F(cid:48)o}r guaranteesd(CS¯)≥0,0,2,respectively.Whiletheboundsfor
0 0 0 0 1 L0 |S| = 0,1 are trivial, the bound for |S| = 2 is significantly
fforormL0{Cm,e.ss.a.g,eCs }ofblyensgetthtinkg0Ce(cid:48)ac=h c(cid:80)anmkb0e readily oCbt.ained lower than d(CS¯)=12 achieved in Example 1.
The g1eneratorLmatrix GGG of Cm, (cid:96) =(cid:96)1=,(.m.−.,1)Lk,0+c1ons(cid:96)ists of
(cid:96) (cid:96)
a single vector ggg ∈ Fn, and the generator GGG ∈ FL×n of V. BINARYCODESFORINFORMEDRECEIVERS
(cid:96) q q
C =(cid:80)L(cid:96)=1C(cid:96) consists of rowsggg1,...,gggL. The generatorGGGS¯ In this section we construct and identify several families of
of CS¯ is a submatrix of GGG composed of the rows ggg(cid:96), (cid:96) ∈ S¯. binary ECCIRs using cyclic codes and the MDSIR codes of
The following lemma characterizes the generator matrices GGG Section IV.
of MDSIR codes for informed receivers when k =1.
Lemma 3. The matrix GGG ∈ FL×n is the generator of an A. Code Concatenation
q
MDSIR code if and only if every square submatrix of GGG is
Binary ECCIRs can be obtained from the MDSIR codes
nonsingular.
of Section IV by concatenating them with a binary inner
Proof: We know that CS¯ is MDS, i.e., d(CS¯) = n − code. Let q =2k and let {C1(out),...,CL(out)} be a length n(out)
|S¯| + 1, if and only if every |S¯|×|S¯| submatrix of GGGS¯ is MDSIR code for L symbols over Fq, i.e., each C(cid:96)(out) is of
nonsingular.Equivalently,CS¯isMDSifandonlyifanysquare dimension 1 over Fq, or equivalently, dimension k over F2.
submatrix ofGGG obtained by selecting the rows corresponding Each of the n(out) F -symbols of the outer MDSIR code
2k
to S¯ and any |S¯| columns is nonsingular. By letting S vary C(out) = (cid:80)L C(out) is linearly mapped to a length k binary
(cid:96)=1 (cid:96)
over all subsets of {1,...,L} we arrive at the statement of vectorandthenencodedbyan[n(in),k,d(in)]binaryinnercode.
the lemma. The resulting binary ECCIR C ,...,C encodes L binary
1 L
Matriceswitheverysquaresubmatrixbeingnonsingularare messagesofsizek eachintoalengthn=n(out)n(in) codeword
known to be related to the generator matrices of (traditional) over F . Suppose a receiver Rx has prior knowledge of
2 S
MDS codes [15]. Let AAA=[III|GGG] be the systematic generator the messages with indices in S (cid:40) {1,...,L}. The effective
matrix of an [n+L,L] linear code over Fq. Then we have binary code CS¯ ⊂ Fn2 at this receiver is the concatenation of
4
CS¯(out) =(cid:80)(cid:96)∈S¯C(cid:96)(out) and the [n(in),k,d(in)] binary inner code. TABLEI
Since d(C(out))=n(out)−|S¯|+1, we have BINARYECCIRSOFSECTIONV-B
S¯
d(CS¯)≥d(in)(cid:0)n(out)−|S¯|+1(cid:1). (1) Irreducible ComponentCodes SumCode
CyclicCode ofECCIR
The outer MDSIR code ensures that the lower bound on dis- C(in) C1 andC2 C =C1+C2
tance improves with the amount of side information available [9,6,2] [18,6,6](6) [18,12,2](4)
at the receiver. [17,8,6] [34,8,14](14) [34,16,6](8-9)
[21,6,8] [42,6,20](20) [42,12,8](15-16)
Example 3. Consider L = 2 binary messages of size k = 3
[39,12,12] [78,12,32](32-33) [78,24,12](22-26)
each. Let the outer MDSIR code be of length n(out) =3 over
F =F .Since2k >n(out)+Lsuchacodeexistsandcanbe [41,20,10] [82,20,26](26-30) [82,40,10](16-20)
2k 8 [55,20,16] [110,20,40](40-44) [110,40,16](24-32)
constructed from a generalized RS code (Section IV). Let the
[65,12,26] [130,12,56](56-60) [130,24,26](45-51)
binary inner code be [n(in),k,d(in)] = [7,3,4]. The resulting
binary ECCIR {C ,C } has length n=21. From (1), C , C
and C =C +C 1hav2e dimensions 3, 3 and 6, and mini1mum2 terms of minimum distance, and C1+C2 has at least half the
1 2
distancesatleast12,12and8,respectively.Sinced∗(21,3)= minimum distance of the best known code.
12 and d∗(21,6)=8, we conclude that d(C )=d(C )=12
1 2
and d(C +C )=8, i.e., all three codes possess the optimum C. Codes for L=2 using primitive irreducible cyclic codes
1 2
minimum Hamming distance. We now design ECCIRs {C ,C } using irreducible binary
1 2
cyclic codes C and C of primitive length n = 2m − 1,
1 2
B. Code Concatenation using Piret’s method m ≥ 3. Consider the cyclotomic cosets of 1 and 3 in Zn,
C ={1,2,...,2m−1}andC ={3,6,...,3·2m−1 mod n}.
Inthissubsectionweconsiderthespecificcaseofcodecon- 1 3
Let C and C be cyclic codes with non-zeroes C and
catenation where L=n(out) =2 and the [n(in),k,d(in)] binary 1 2 1
inner code C(in) is an irreducible cyclic code. It is known that C3, respectively. We observe that C1 and C2 are linearly
any[n(in),k]irreduciblecycliccode(asanidealinF [x]/(xn− independent (since C1 and C3 are non-intersecting) and they
2
1)) is isomorphic to the finite field F [15, p. 225]. encode k =m message bits each (since |C1|=|C3|=m).
When L=n(out) =2, the binary2kECCIR {C ,C } has We know that C1 is a [2m−1,m,2m−1] simplex code. If
1 2 gcd(3,2m−1)=1, we have C =µ (C ) (from Lemma 2),
length n=2n(in). The lower bound (1) guarantees that 1 3 2
d(C ),d(C ) ≥ 2d(in). By exploiting a known technique due and hence, C2 is a [2m−1,m,2m−1] code as well. If
1 2 gcd(3,2m − 1) (cid:54)= 1, the irreducibility property of C can
to Piret [17], we can optimize the outer code and guarantee 2
thatd(C ),d(C )islargerthan2d(in).Todoso,werestrictthe still be used to compute d(C2) using efficient algorithms,
1 2
2×2generatormatrixoftheoutercodeC(out) =C(out)+C(out) for details see [15, Ch. 2], [19] and references therein. For
1 2 instance, the value of d(C ) for m = 4, 6 and 8, i.e., for
to the form 2
(cid:18)1 β(cid:19) n = 15, 63 and 255 can be computed to be 6, 24 and 120,
GGG= ,
β 1 respectively.
To analyze the minimum distance of C = C + C , we
1 2
where β (cid:54)=1, and we use a known finite field isomorphism observe that the equivalent code µ (C) (with non-zeroes
ϕ:F →C(in) [15], [17] to concatenate the outer MDSIR −1
2k C ∪C )isthedualofthedouble-errorcorrectingprimitive
−1 −3
code with the inner irreducible cyclic code. Thus, the two
BCH code (with zeroes C ∪ C ). Applying the Carlitz-
component codes C1 and C2 of the binary ECCIR are Uchiyama lower bound, we k1now t3hat d(C) is even and
{(ϕ(a),ϕ(βa)) |a∈F2k} and {(ϕ(βa),ϕ(a)) |a∈F2k}, d(C)=d(µ (C))≥2m−1−2m/2.
−1
respectively. Note that C1 and C2 are equivalent up to co- The ECCIRs {C ,C } of this subsection for m ≤ 8 are
1 2
ordinate permutation. Piret [17] considers codes of the same
summarized in Table II. The table also shows the values of
structure as C1 and uses a search to find the value of β that (or the best known bounds on) d∗(n,k) and d∗(n,2k), which
maximizes d(C ); see also [15, p. 588]. The optimal values
1 correspond to the length and dimension of the components
of β (cid:54)= 1 and the resulting d(C1) corresponding to several codes C ,C and the sum code C = C +C , respectively.
binary non-primitive (n(in) (cid:54)=2m−1) irreducible cyclic codes 1 2 1 2
We observe that most of the codes in Table II equal the best
C(in)areavailablein[17],[18].Theparametersoftheresulting
known codes in terms of minimum distance.
codesC ,C andC +C areshowninTableI.Theminimum
1 2 1 2
distancesd∗(2n(in),k)andd∗(2n(in),2k)oftheoptimalbinary
linear codes with the same length and dimension as C , D. Codes for L=2 from quadratic residue codes
(cid:96)
(cid:96) = 1,2, and C +C are shown in parentheses. If the exact The ECCIRs of Section V-C are of low rate since the
1 2
value of d∗ is not known best available bounds are given. component codes C and C are irreducible cyclic codes. In
1 2
Note that since GGG is a 2 × 2 invertible matrix, we have this subsection we identify a class of high rate ECCIRs for
C +C = (cid:8)(aaa,bbb)|aaa,bbb∈C(in)(cid:9). Consequently, C +C is L = 2 messages where each component code is a quadratic
1 2 1 2
a [2n(in),2k,d(in)] code. For every ECCIR presented in Table I residue(QR)codeandthesumcodeisthe[n,n−1,2]single-
we observe that C and C equal the best known code in parity check code. Binary QR codes are a family of cyclic
1 2
5
TABLEII TABLEIII
BINARYECCIRSFORL=2MESSAGESWITHk=mANDn=2m−1 BINARYECCIRSWITH(cid:80)3(cid:96)=1C(cid:96)=[n,n−1,2]CODE
n k d(C1) d(C2) d∗(n,k) Loownerd(bCou)nd d∗(n,2k) n k C(cid:96) C(cid:96)1+C(cid:96)2
(cid:96)=1,2,3 1≤(cid:96)1<(cid:96)2≤3
7 3 4 4 4 2 2
31 10 [31,10,10](12) [31,20,6](6)
15 4 8 6 8 4 4
31 5 16 16 16 12 12 43 14 [43,14,14](14) [43,28,6](6-7)
63 6 32 24 32 24 24-26 109 36 [109,36,24](26-34) [109,72,10](12-16)
127 7 64 64 64 54 56 127 42 [127,42,28](32-40) [127,84,14](14-18)
255 8 128 120 128 112 112-120
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