on the concept of code-isomorphy

Journal o f Geometry Vol. 57 (1996)

0047-2468/96/020063-0751.50+0.20/0 (c) Birkhauser Verlag, Basel

ON THE CONCEPT OF CODE-ISOMORPHY

In memoriam Giuseppe Tallini

I o a n a C o n s t a n t i n e s c u a n d W e r n e r Heise

Let F be a finite set of cardinality IF] =:q >__ 2, n _> 1 an integer and 0 : FnxFn--*lgo the HAMMING metric. A code isomorphism C --* D between two block codes C,D C_ F n is defined as an isometry which can be extended to an isometry of the whole space F n. Any permutation ~r E ~ of the positions canonically induces a so-called equivalence map fr E AutF n ; any system ~ := (~1,~2,... ,~n) of n permutations of the character set F induces a so-called configuration ~ E AutF n. The group AutF n of all isometries of F n turns out to be semidirect product of the configuration group with the symmetric group of degree n. The codeword estimating failure probability of a maximum likelihood codeword estimator for a q-nary symmetric channel does not depend on the transmitted codeword, if the automorphism group of the code acts transitively on the set of codewords. When using a systematic (n,k)-encoder, the symbol decoding failure probability does not depend on the transmitted symbol or on the time of transmission if the configuration group and the automorphism group act transitively on the set of codewords resp. on the set of the k information positions.

1. C O D E S

Let q_> 2 be a n in t ege r a n d let F be a q-nary character set, i .e. a f in i te set

F - { 0 , 1 , . . . } of c a r d i n a t i t y IFI = q. W e cons ide r a c o m m u n i c a t i o n s y s t e m w i t h a

q-nary symmetric channel of error probability p < 1 - ~- (qSCp), i.e. a "stochast ic

m a p p i n g " X : F ~ F w h i c h t r a n s f o r m s a s y m b o l a E Y in to a s y m b o l fl E F w i t h

a - p r i o r i - p r o b a b i l i t y P(~I c~) I 1 ~-p, if a : "---- if a ~ ~ ' Lq-I

The messages to be transmitted are selected with equal probabilities from a finite

non-empty set ~. Using a (deterministic) injective coding map C: ~--+ F n (with a

prefixed integer n), the encoder transforms the selected message rn E~g'/into a codeword c := C ( m ) f r o m the (q-nary block-) code C : = C ( . # t ' ) ___ F n (of length n).

64 Constantinescu and Heise

In the case ~/-/= F k, k E N, systematic encoders C : F k -+ F n are recommended: Each

information word m E F k will be encoded into a codeword C(m) = ( m , r ( m ) ) E F n

by annexing a control sequence r (m) E F r of length r := n - k to m . The resulting

code C := C(F k) is then called a systematic (n,k)-eode.

No mat ter which code is used, after the input of a codeword c E C, the channel

applies the stochastic mapping X to every component of c and emits a possibly

distorted channel word w := X(c) := XXXX . . . x x ( c ) E F n. The (maximum likelihood)

codeword estimator (MLCE) transforms this channel word into a codeword c~E C,

whose probability p(w[c ' ) is maximal; for reasons of homogeneity, we assume the

MLCE to be balanced, i.e. in cases in which there is no uniqueness, the MLCE

chooses the candidate c ' E C at random. Finally, the decoder sends the message

c - i ( c ') E J / t to the message sink.

The appropriate tool for the description of error-correcting codes for the q-nary

symmetric channel is the HAMMING-metric ~ :FnxFn---+ No, which associates to

every pair ( x , y ) E F n x F n the number of positions, in which the words x and y

have distinct components. The a-priori-probability that the qSCp will change an

input word c E F n into an output word w E F n is

p ( w l c ) =

In order to describe the MLCE, for every word w E F n we define its candidate sei

~(w) = ( c ' e C ; V c e C : e(c,w) _ 0(c ' ,w)}.

The MLCE is a stochastic mapping e : F n ~ C which maps a channel word w E F n

into a codeword c ' E d'(w) with probability 1

The arithmetical mean of the codeword estimating failure probabilities _ v " I(c}n (w)l

..= 1

evaluated over all codewords c E C serves as a yardstick for the reliability of the

communication system. Observe that l{ c} N ~(w) t = [ 1, if c ~ r

Consider a communication system for which the messages are the information words

of length k with components from the character set F and in which a systematic

encoder C :F k ~ F n and a MLCE are integrated. The user of such a communication

system is more concerned about the probability that a single symbol a E F emitted

by the information source at a certain instant will not survive the data processing and

reach the sink as another symbol than about the probability that a whole message,

i. e. an information word, will be distorted. For any position i E { 1 , 2 , . . . ,k} and any

symbol a E F, we denote by

Ci,a := { C l C 2 " " % % + l " ' c n E C ; c i=c~}

Constantinescu and Heise 65

the a-derivation of the systematic (n,k)-code C in position i , which (after deletion

of the i th component a in every codeword) itself is a systematic ( n - l , k - 1 ) - c o d e .

The symbol decoding failure probability

is the measure for the chance that a symbol a E F which functions as the i th

component c i = a in a message m := c lc2 . . , c k E F k - or equivalent]y as the i th

component c i = a in the codeword c := C(m) = c lc2 . . . Ck Ck + 1 . . . c n C C - will

reach the destination sink altered. Evidently we always have pf~z(c,i) <_ p/~(c).

Since any non-codeword w E F n ~ c can be regarded as a potentially disturbed

codeword, our definition of a code as a non-empty subset C C_ F n should be stated

more precisely :

A code is a triplet (F n, ~ ,C) , where C is a non-empty subspace of the metric space (F n Q).

2. C O D E - I S O M O R P H I S M S

Accepting this formal definition, one has to define an isomorphism between two codes

C,D C_ F n as an isometry C ~ D which can be extended to an isometry of F n. We

denote by ' I s o C ' the group of all isometrics of a code C C_ F n onto itself; thus the

automorphism group AutC of a code C _C F n is isomorphic to the group I s o c F n of

all mappings ~ E I soF n = AutF n leaving C invariant factorized by the group of

all isometrics of f n which leave C pointwise fixed.

EXAMPLE. The ternary codes C := {000,011,022} and D := {000,011,110} are both

equidistant and hence isometric. The word 010 has distance I from every codeword of

C, but there is no word in {0,1,2} 3 having distance 1 from every word of D, thus the

codes C and D are not isomorphic. In geometrical language this sounds paradoxical -

in the metric space ({0,1,2}3, t~) the equilateral triangles C and D of side-length 2

are not congruent! El

Any permutation 7r E ~n induces an isometry on F n, the so-called equivalence map

: Y ~ - ~ F ~ ; x l x 2 . . . a s ~ x ~ - l ( 1 ) % r - l ( 2 ) - �9 �9 z ~ - l ( ~ ) .

We denote by 'EquaF n' the group of all equivalence maps which leave a code

C C F n invariant. The group of restrictions of the mappings of Eq u v F n onto the

code C will be denoted by ' Equ C'.

The group ConfF n < IsoF n of all n-tuples ~ := (~1 ,~2 , . . . ,~n) of permutations

of the character set F acts on F n as the group of so-called configurations

;


For any code CC_Fn , the notations 'Confc F n ' and ' C o n f C ' are self-explanatory.

If q is a prime power, then we identify the character set F with the GALOIS field IFq

and the cartesian product F n with the vector space Vn(q).

In a purely formal analogy with a theorem of WITT [1; p. 121f] on the extendability of

isometries between linear subspaces of metric vector spaces the monomial theorem ([3],[4], [5], [6;p. 295ff.]) states that any linear isometry between two linear codes

C , D <__ Vn(q) can be extended into a monomial transformation of Vn(q) , i.e. the

product of a non singular diagonal matrix and a permutation matrix in GLn(q). A general 'monomial theorem' does not exist for nonlinear codes nor for nonlinear isometries of linear codes. However, the following characterization of the group iso f n of all isometries of the whole space Y n can be useful to show that. a certain

group of isometries of a code is the full automorphism group of this code.

THEOREM 1. Any isometry of F n is the product of a configuration and an equivalence map.

Proof . For any word a = X l X 2 . . . x n E F n, we define its HAMMING-weight

~(x) = l{i ;xi r 0}1 as its HAMMING-distance Q(x,O) from the all zero word 0 "= 0 0 . . . 0 E F n, For any

symbol c~ E F \ { 0 } and any subscript i E {1,2, . . . ,n} the word ea, i E F n (which has the character c~ in its i th position and zeroes elsewhere) has HAMMING-weight 1.

Apart from these n - ( q - 1 ) words there are no other words in F n of weight 1.

Let ~ E Iso F n be an isometry. We may assume that ~ fixes the all zero word,

because otherwise we could choose n permutations ~i E ~fF, i -- 1,2~... ,n inter- changing the i th component of the word ~(0) with 0 and multiply ~ by the

configuration ~ := @ 1 ~ 2 , . . . ,;~n)'

For any subscript i E {1 ,2 , . . . ,n} we have ~/(~(el,i) ) = 1; thus there always exists a

symbol Q E F ~ { 0 } and a subscript ~riE{1,2, . . . ,n} with ~(el , i ) = e~,~ri. The

existence of two distinct subscripts i , j E {1,2 , . . . ,n} with 7r i =~rj would lead

to the contradiction 1 > g(ee~,ri,ecjrj) = g(~(el , i ) ,~(el , j ) ) = 2. Thus the mapping

{ 1 , 2 , . ,n} {1,2, . . . ,n} ;i is a permutation, e Let i E (1,2, . . . ,n} be a subscript. For any symbol o~ E F~.{0} we have

~(~(e~, i) ) = 1; thus there always exists a symbol ei(c~ ) E Y ~ ( O } and a subscript

t i (a ) E {1,2 , . . . ,n} with ~p(ea,i) -- eEi(a),ti(a ). We have ti(1) = ~r(i). If a =~ 1 then

1 = ~(~(e l , i ) ,~(ea , i ) ) = 0(e~,r(i ),ee~(a),ti(a)), hence t i (a ) = ~r(i). For any two distinct

symbols a , f l e F \ { 0 } , we have 1 = ~(~(ea, i ) ,~(e~, i ) ) = e(e~,(a),~r(i),ee~(~),~(i)) and

thus ei(~ ) ~ ei(/3); the mapping e i : F ~ { 0 } ~ F ~ { 0 } ; a H ei(o~), extended to P

by setting ei(0) ,= 0, is bijective, e i E "~F" We abbreviate ~i "= eTr-~(i)"


Let x l , x 2 E F n be two words with the following property. For every position at

which the components of x 1 and x 2 are distinct, one of these components is zero.

We define the OR-word x 1 v x 2 of x 1 and x 2 as the word which coincides with x I

and x 2 at all positions where x 1 and x 2 coincide and which has the non zero com-

ponents of x 1 or x 2 at those positions where x 1 and x 2 do not coincide.

The isometry

:= : --+ ; xlx2.-. % --- %(%-i(n)) has common values with W for all arguments x E F n of weight 7(x) _< I. Assume

there would be a word x c F n of minimal HAMMING-weight 9 with v~(z) v~ ~(x), i.e.

a word x E F n of minimal HAMMINe-weight g which would not be fixed point of the

isometry r :--v~-lo~. This lightweight wrongdoer would possess at least two non

zero components. Let x 1 resp. x 2 be two difffferent words which emerge from x by

exchanging one of the non zero components of x with 0. We conclude x -- x 1 v x 2.

Because of the minimMity of g, we have %b(xl)--x 1 and r x 2. From

7(~b(x)) -- g, 7(Xl) -- 7(x2) -- 9 - 1, Q(Xl,X2) -- 2 and ~(%b(x),xl) -- O(r = 1, i t f o l l o w s t h a t = = a C o n t r a d i c t i o n to our a s s u m p t i o n a

The group ConfF n is a normal subgroup of IsoFn; indeed, if s E ConfF n and

v? E EquFn, then ~?-1o ~o # (x lx2 . . . Xn ) __ ~(1)(xl ) ~r(2)(x2).. . ~r(n)(xn ) for every word X l X 2 . . . x n C F n. Thus IsoF n is the semidirect product of ConfF n and

EquF n. In the language of permutation groups, the isometric group IsoF n is the

wreath product of the symmetric groups ~n and J r '

EXAMPLE. In 1951 F. L. BAUER [2] took out a patent on a coding map for the binary

extended (8,4)-HAMMING-code. Let F be the set of the two bits 0 and [_. For any

word x with components in F, we denote by x* its complement, i.e. the word

emerging from x, after substituting each component by the other bit.

BAUER'S encoder: B : F 4 - + F S ' x H : = ~ (x'x) 'ifT(x) -= ~ m~ 2 ' [ (x,x*), if 7(x) 1 rood 2'

The resulting code B := B ( F 4) C_ F s is a binary systematic (8,4)-code; if we identify

the character set F with the prime field ~E2, then B C_ Vs(2 ) is a linear code. For

any codeword b E B, all 14 codewords c E B \ { b , b * } have distance L)(b,e) = 4

from b. Hence IsoB can be interpreted as the wreath product of the symmetric

groups ~s and S~2 and has order IlsoBI = 10321920; as abstract groups, IsoB and I soF a are isomorphic. The group of all permutations Tr E ~8, which induce equi-

valence maps # C EquBF 8, is generated by the permutations (1 5)(2 6), (15)(3 7),

(1 5)(4 8), (1 3 2)(5 7 6), (1 2)(5 6) and (1 2 7 3 5 4 6). The group EquBF 8 acts faith-

fully on B, thus ]EquB I = 1344. The code B does not admit other linear isometries.


The group ConfB consists of the 16 translations B ~ B ; x ~ b + x with a code-

word b E B. Hence, AutB is a subgroup of IsoB of index [IsoB :AutB] = 480. Cl

3. C O D E P E R F O R M A N C E

We consider again a communication system with an information source ~ (from

which the single messages are selected with equal probabilities) , a symmetric q-nary

channel X : Y ~ F of error probability p, an encoder C : ~ / t ' ~ F n, a balanced

maximum likelihood codeword estimator e :Fn--* C := C(..~) and the decoder c - i : c - , .//{.

An old country saying states that among optimal (with respect to the error correcting

capability) codes with appropriate parameters, there are often linear codes. Indeed, it

seems reasonable, that the codewords of a good code should be equally protected

against the evil channel activities. Equal protection means homogeneity, homogeneity

means transitivity of the automorphism group:

THEOREM 2. I f Aut C acts transitively on the set C of codewords, then for any two

codewords c , c ' E C the estimating failure probabilities coincide, plait(c) = pf~iz(e')-

P roof . For any isometry ~ E IsocP n, any codeword c E C and any word w E F n

we have p(w]c) = p(~o(w)[(fl(c)) and (p(-C(w)) = ~((p(w)). El

For practically all good codes, the configuration group ConfC acts transitively on the

set of codewords; this is especially true for all linear codes C <_ Vn(q). For any code-

word c E C, the translation C ~ C ; a ~ c + x is a configuration.

To design a systematic encoder C : F k ~ F n one should equally protect all symbols

of the information source F against the channel noise. The next theorem tells us,

how such a homogeneity can be achieved:

THEOREM 3. I f AutC acts transitively on the set { 1 , 2 , . . . , k } , i.e. if for any two positions i, i ' , 1 <_ i, i ' < k, there is an configuration ~ and an equivalence map ~r with ~o'~ E I s o c F n and Tr(i) = i ~, and if ConfC acts transitively on the set of codewords, then for any two codewords c ,c l E C and any two positions i, i / E {1,2, . . . ,k} we

have pfo (c,i) =

Proof . Let ~ = koT? E IsovF n be an isometry of F n which leaves C invariant

with a configuration ~ := (~ci,~2,... ,~n) and a permutation ~r E 5z n. (We explicitely do not forbid that ~r may be the identity!) Then for every information position

i E {1 ,2 , . . . ,k) we have ~(Ci,ci ) = CTr(i),ag(i)(ci ) and ~(~(w)) = ~(qa(w)). Since

= for every word e F , w e get =


The homogeneity conditions of this theorem are fulfilled for cyclic linear codes, and

they are hereditary to linear extensions of linear codes.

4. A C K N O W L E D G E M E N T S

We are grateful to Dr. THOMAS HONOLD and Dr. MICHAEL KAPLAN for their patient

willingness to discuss time and again the ABC of coding theory. We express our special appreciation to Professor Dr. Dr. h. c. mult. FRIEDRICH LUDWIG BAUER, who told us about his participation in the very beginning of coding theory. We also owe thanks to Professor Dr. RAFFAEL ARTZY and Professor Dr. JOSEPH ZAKS for giving us the opportunity to present this paper at the Seventh International Conference on Geometry

in Nahsholim/Israel. Professor Dr. SANDRA HAYES supported us in our fight for correct use of English prepositions, thank you, Sandra!

R E F E R E N C E S

[1] ARTIN, E., Geometric Algebra. Wiley-Interscience, New York 1957

[2] BAUER, F. L., Verfahren zur Sicherung der ~/bertragung yon Nachrichtenimpulsgruppen gegeni~ber StSrungen. Patentschrift, DBP 892 767 (Anmeldung am 2t. Januar 1951)

[3] BOGART, I4., D. GOLDBERG and J. GORDON, An elementary proof of the MacWiIliams Theorem on equivalence of codes. Inf. and Contr. 37 (1978) 19-22

[4] FILIP, P. and W. HEISE, Monomial code-isomorphisms. Ann. Discr. Math. 30 (1986) 217-224

[5] GOLDBERG, D., A generalized weight for linear codes and a Witt-MacWilliams Theorem. J. Comb. Th. A 29 (1980) 363-367

[6] HEISE, W. und P. QUATTROCCHI, Informations- und Codierungstheorie. 3. Aufl., Springer-Verlag, Berlin-Heidelberg-New York-Tokio 1995

Mathematisches Institut Technische Universits D-80290 M<inchen Germany

constant <@ mathematik.tu-muenchen, de heise @ mathematik.tu-muenchen, de

Eingegangen am 24. Februar 1995

on the concept of code-isomorphy

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