Table Of ContentLecture Notes ni
Computer Science
Edited yb .G Goos and .J Hartmanis
228
deilppA Algebra, scimhtiroglA
dna gnitcerroC-rorrE Codes
dn2 lanoitanretnI AAECC-2 Conference,
,esuoluoT October France, 1-5, 4891
sgnideecorP
detidE yb Alain iloP
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nilreB grebledieH NewYork London siraP oykoT
Editorial Board
D. Barstow W. Brauer .P Brinch Hansen D. Gries D. Luckham
C. Moler A. Pnueli G. Seegmi311er .J Stoer N. Wirth
Editor
Alain Poll
AAECC/LSI Lab., Universit~ .P Sabatier
118, route de Narbonne, 31062 Toulouse C~dex, France
CR Subject Classifications (1985): B.4.5, G.2.0
ISBN 3-540-16767-6 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-16767-6 Springer-Verlag New York Berlin Heidelberg
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ECAFERP
The Internation~ Colloquium no Applied Algebra and Error Correcting Codes saw
born in Toulouse (France) in June 1983.
The acts of I-CCEAA ~e publ~hed in Discrete Mathematics (vol 65 n~2-3, 0ct. 85).
The acts of 2-CCEAA are contained in this volume.
morF 84 talks, ew have selected 32 accepted papers, after a (time consuming)
system of multiple reviews. I thank those referees ohw agreed to contribute to the
obtained result.
I also thank :
• .rM .A Oisel and CII-HBull for their financial support,
• ~r. .M Comb~nous Scientific Director of ,*SRNC for CNRS's financi~ suppo~Z,
.rM .A Dargent, Director of SENC ~* Informatic Center, for allowing us the use of
the compeers before and ~'ng the conference,
• The LSI labor~ory and University .P Sabatier for their financial support.
sA one knows, digitalized data are becoming increasingly important, pa~cular-
ly for transfusions.
For satellite transmissions, the S~SCC (Consultative Committee for D~a Space System)
had proposed a coding system for international transmissions (see : fin~ report of
contract SENC/CCEAA o n 84/5417, 1985 (210 pages)).
Also, the targ~ of ECAR project is to define and realize a Broadband-IBC european
n~twork with security/privacy (cryptography) and reliability ( er~or-co~ecting co-
des). CCEAA lab. is a participant for the definition phase (in group n°2015).
sA digitalized data are b~ng more and more used for images~speech~flies trans-
missions, theoretical tools and practical developments are necessary {for finite al-
gebr~c structures and for comple~y analyses).
In parCLcular, decomposition of algebras is an interesting topic because it is used
for problems involved with complex/~ (see J. Heintz/J. Morgenst~nl, for ~czu~tsnoc
tive resu~ no idempotents, et~&~J~vi~tlum codes (see : .A Poll, .H Imai, A. Poli/
.C Rigoni), for DFT's problems (see : T. Beth). Many other p~ticular aspects of re-
SRNC Centre National de la Recherche Scientifique
~ SENC Centre Natio~ d'Etudes Spatiales 81( Av. NILEB - 31055 ESUOLUOT lxed~C
iv
search are dev~oped in t~is book. Cov~n~_ra__~__s .G( Cohen/N.J.A. Sloane/A.C. Lob-
stun, H.F. Mattson Jr., .L Huguet/M. G*~era), §o~cti_o~s~a_~o~or~_~so~_cod~
(J.A. Thiong-Ly, J~L. Dornstet~er, D.A~ Leonard/C.A. Rodg~t, B. Cou~eau/J. Goulet},
~do§_fo__jt§§~_~a__~_§itca@~ (M.C. Gennero, G.L. Feng/K.K. Tzeng), __~a__~_o~{lo~ {P.Piret,
.D iugiez, .O Moreno de Ayala), aE~li§dal~§bra IH.M. M611er/F. Mora, .L Beneteau/
I. Lacaze, A. Astie-Vidal/J. Chiffl~}, c r~t{~@a~h{ (P. Camion), @~E_~_o~§b_m~_
(J. Calmer, J. C~lmet/~. B~gman}.
" CCEAA Conferences essentkally deal with Applied Algebra, J£gorkthmic and Error
Correcting Cod~.
The future scheduled CCEAA conf@¢ences are :
3-CCEAA (1985, Grenoble (F), Prof. J. C~m~)
•
4-CCEAA (1986, Karlsruhe (DJ, Prof. Dr. T. Beth)
•
5-CCEAA (1987, Barc~ona (SP), Dr. .L Huguet)
•
6-CCEAA (1988, Pica (I], Prof. A. ~iola)
.
7-CCEAA (1989, Toulouse (FI, Prof. A. Poli)
•
8-CCEAA (1990, amahokoY (J), Prof. .H Imai)
•
eW hope that CCEAA. Conferences, and pa~v%icularly t@~ Lecture Notes volume, will
contribute to the important dev~opment of data transmissions.
Finally, a thank you to participants, authors, and also to Miss S. Watson
ISpring~ Verlag Computer Science Edltori~l for h~t pc~ence and very kind help.
A particular thanks to the s~ies editors ohw have accepted t~ publkcation.
yaM 1986 Alain ILOP
STNETNOC
J. HEINTZ, J. NRETSNEGROM I
nO" associative algebras of minimal rank"
A. lLOP 52
"Construction of primitive idempotents for n variable codes"
.H IMAI 63
"Multivariate polynomials in coding theory"
A. POLl, .C INOGIR 16
"Enumeration of self dual k2 circulant codes"
T. HTEB 17
"Codes, groups dna invariants"
G.D. ,NEHOC A.C. ,NIETSBOL N.J.A. ENAOLS 97
nO" a conjecture concerning coverings of space" gnimmaH
H.F. NOSTTAM Jr. 09
nA" improved upper bound no covering radius"
L. ,TEUGUH .M AREIRG 701
"Association schemes dna difference sets defined no owt weight codes"
J.A. YL-GNOIHT 211
"Automorphisms of two families of extended non binary cyclic appoG codes"
J.L. RETTETSNROD 221
emoS" quasi-perfect cyclic codes"
D.A. ,DRANOEL C.A. REGDOR 831
"Explicit Kerdock codes over GF(2)"
.B ,UAETRUOC J. TELUOG 631
"Une classe ed codes 2-correcteurs adapt#s aux syst~mes d'information
format,s"
M.C. ORENNEG 541
CCITSUOL" simulation software : experimental results of coding systems"
G.L. ,GNEF K.K. GNEZT 451
nA" algorithm of complete decoding of double-error-correcting appoG
codes"
.P TERIP 161
nO" the number of divisors of a polynomial over GF(2)"
.D ZEIGUL 961
"Multivariate polynomial factoring dna detection of true factors"
.O ONEROM ED ALAYA 871
"Discriminants dna the irreducibility of a class of polynomials"
H.M. ,RELLOM F. AROM 281
"Computational aspects of reduction strategies to construct resolutions
of monomialideals"
Vi
L. ,UAETENEB J. EZACAL 198
"Designs arising from symplectic geometry"
.A ,LADlV-EITSA J. TELFFIHC 206
"Distance-transitive graphs dna the problem of maximal subgroups of
symmetric groups"
.P NOIMAC 512
naC" a fast signature emehcs without secret yek eb secure "?
J. TEMLAC 242
"Manipulation of recurrence relations in computer algebra"
J. ,TEMLAC .M NAMGREB 352
emoS" design principles for a mathematical knowledge representation
metsys : a wen approach to scientific calculation"
ON ASSOCIATIVE ALGEBRAS OF MINIMAL RANK
Joos Heintz )I and Jacques Morgenstern )2
)I Consejo Nacional de Investigaciones Cientlficas y T6enicas (CONICET)
Universidad Nacional de La Plata
La Plata, Provincia Buenos Aires, Argentina
and
Johann Wolfgang Goethe - Universit~t, Fachbereich Mathematik
Robert Mayer - Strasse 0 ] - 6
D- 6000 Frankfurt/Main, FRG (mailing address)
)2 Universit@ de Nice, Inst~tut des Math~natiques et Sciences Physiques
Parc Valrose
F - 06034 Nice Cedex, France
and
,AI~qhI Sophia Antipolis
F- 06560 Valbonne, France
I. Introduction
In the sequel let k be a field and A an associative k-algebra with
unity, of finite dimension over k. We denote the radical of A, the
maximal (two-sided) nilpotent ideal contained in A, by tad A .
A quadratic algorithm (for A) is a finite family
* *
= ((Up,Vp,Wp) E (A xA) x(A ×A) × A ; p= I ..... R)
R
satisfying xy = Z Up(X,y) Vp(X,y) pW , V x,y 6A
p=]
,
(Here A) (A × denotes the dual space of the k-vector space A xA. )
Special cases of quadratic algorithms are the bilinear algorithms
(for A) which have the form S = ((Up,Vp,Wp) 6A × A × A ; p= I .... ,R)
with
R
)I( x y = I P =I Up(X) Vp(y) Wp , V x,y CA .
(Note that t := Z u D® v Q® Wp 6 A ®~A ®~A is the tensor of the multipli-
cation of the algebra A and hence doesn't depend on the particular
algorithm $. )
For = 8 ((uQ,vQ,wp); p= I, .... R) a quadratic or bilinear algorithm we
call L(~) := R the complexity of 8 • We define the following invari-
ants of A :
L(A) := min { L($); $ quadratic algorithm for A , } the complexity of A
and
R(A) := min { L(~); ~ bilinear algorithm for A } , the rank of A.
It is well known ([17]) that L(A) , the complexity of A , can be inter-
preted as the computational complexity of multiplying two generic ele-
ments of A.
Furthermore, we have L(A) ~ R(A) ~ 2 L(A)
So, for asymptotic complexity considerations, L and R are equivalent
notions. This fact has widely been used for the construction of fast
matrix multiplication algorithms (compare e.g. [16],[5],[15],[7]).
Fast matrix multiplication and fast convolution algorithms are at the
origin of the consideration of bilinear algorithms (compare e.g. [16],
[17],[18],[19]).
The rank of an algebra appears to be closer related to the structure
of A than its complexity. For this reason we focus our attention on
the rank of algebras.
The starting point of our considerations is the following lower bound
result for the complexity of associative algebras.
Theorem I ([I]) L(A) a 2 dimkA - # M(A) ,
where M(A) m := { ; m maximal two-sided ideal of A } and # M(A) is
its cardinality.
(In the case of A := MN(k) , the algebra of N×N matrices over k , the
result is due to [14].)
We will use the following notions :
Definition I We say
)i( the complexity of A is minimal (L(A) minimal)
iff L(A) : 2 dimkA - # M(A) ;
(ii) the rank of A is minimal (R(A) minimal)
iff R(A) = 2 dimkA - # M(A)
Observations I
.I We conjecture L(A) minimal iff R(A) minimal. This has been shown
for A a division algebra ([8],[3]). In general, we only know that
R(A) minimal implies L(A) minimal.
.2 M2(k) is of minimal rank ([16]). We conjecture that M~k) is the
only matrix algebra of minimal rank.
3. Let k be infinite, X an indeterminate over k, and F(X) 6 k[X] .
Then A:= k[X]/(F) is of minimal rank ([19]).
We call the k-algebra A local if A/rad A is a division algebra.
We call A clean if for each maximal two-sided ideal m of A the
k-algebra A/m is a division algebra. This is equivalent to saying
that A/tad A is a finite product of division algebras.
Note that A commutative implies A clean.
An important example of a non commutative clean k-algebra is TN(k) ,
the algebra of the upper triangular N×N matrices over . k
We are considering the following class of R-algebras :
Definition 2 Let A be a clean k-algebra with n := dimkA and such
that A/rad A is a p-fold product of division algebras.
We say that A belongs to the class k M (in symbols: A 6 Mk), if there
exists a pair of bases Z= ~Xl,...,Xn),(yl,...,yn) ) of A which satis-
fies the following properties :
)i( V x y~6 k x + k yz for each I ~v,Z ~n ;
(~) x = ~y for I Sz ~p , and I x ,...,Xp are mutually orthogonal
idempotents of A. Furthermore I = ~ + + ... Xp ,
x y~6 k y~ and x y~6 k x for I ~ ~p , I ~,~ ~n .
A pair E of bases of A which satisfies )i( and (~) is called a multJr
plicative pair (for short: an M-pair) of bases of A .
We remark that our notion of class k M coincides with the one used in
[6] in case of local k-algebras.
It is possible to characterize the class of commutative algebras of mi-
nimal rank over an infinite field k. In the case that k is algebraic-
ally closed, the result can be stated as follows :
Theorem 2 ([]3]) Let k be algebraically closed and A be commutative.
Then the following three conditions are equivalent :
)i( R(A) minimal.
)~( A £ k M
)i~i( The radical of A has the form tad A = ( ~I .... '~m ) , where
~i mj = 0 for I $i#j m S .
Condition (~) is a structural characterization of the class of commu-
tative algebras of minimal rank over k. Our goal is to find such a
structural characterization for any associative algebra of minimal
rank. In this paper, we resolve this problem for the class of clean
k-algebras in case k algebraically closed.
In the statement of our main result (Theorem 3) we use the following
notions and notations :
Let A be a finite dimensional clean k-algebra over an algebraically
closed field k.
For e 6 A we denote by A ~ A the k-vector space of A generated by
the products b ~ a , where a,b 6 A . A w A is the minimal two-sided
ideal of A which contains ~.
Furthermore we write L and R for the following two-sided ideals of A :
L := x 6 { tad A ; x (rad A) = } 0 and R := y [ 6rad A ; (tad A) y = ] 0
We have then
Theorem 3 In case k algebraically closed and A clean the following
three conditions are equivalent :
)i( R(A) minimal.
(~) A E k M
(~) There exist e1,~..,~m 6 tad A such that
tad A = L + A ~IA + ...+ A ~m A : R + A ~]A + ...+ A ~ A m
Here, in particular, the two-sided ideals A ei A are as k-vector
2 f
subspaces of A generated by the products ~i'ei ' .... ~i'''" ' f 6 ~ ,
f
or equivalently: A ~i A = f X 6~ k~ i
Furthermore, ~1 ..... ~m satisfy ~i ~j = 0 for I ~i#j ~ m.
