Marhemar~sche~ [nrdtur der Univers,:iit T;~hip~cn. Auf der 'vlor~e~tetle i~J. ;4r~h TiJha~en i. Federal Repl,hlic o[ Germany Received 12 Deccmbq.r ;978 Revised 2 ,'~pri] i979 A c~}de C is Lal'ed homogeneous if the distances ~f c~lde wยข~rds are equally distributed am(in*: the co ~rdinates It is ~hown that if C is a nontr~via: hernogeneou~ code the distance distribution of the punctured code C ~ and that of ~ determine each o*her, and the rninimmn di~tancts at{C) and talc *) arc related by m(CI m(C ~) * I. If C is a linear code of di:ne;ision at lea~t ~.
rloreove: m(rl nz(C') whet,' C' is a suitable hyperplane .ff C ~ A well known theorem of E. P~ange [2] describes the relation between ~he ~eighl enumerator of a binary code C and the weight enum~ ~atnr of its extension ~;; '~y ar over-all parity check provided C is invariant under a permtdation group ~e;muting trans;tively the coordinates. This theorem has been generalized in val'aus ways for binary linear code , see [1, pp. 232-233]. The aim of this note is to put it into the much more gene-d frame,,vork o +" arbitrary codes over alphabets f" of ~rbitrary finite order q ~>2. ',.'he assumption of a transitive group can be r,e~')l~-ced by a homogeneity condit~ n which is at least ~ormally weaker. Suppose F is an alphabet of finite order q 2. Then let d(x, y) -1 if x 7' y and a (x. x)-0 for x, y s F (0, 1 real numbers). In the set F" of F-wo!ds x-(x,)~,, oJ I:ngth n the Hamming distanci: d can be defined by ,l(x. y)= ~ d(x,, y,) ~,here x=(x,) and y=(y,).
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length n over F is a nonempty subset of F". C i~ c'd, d trivial if IcI := 1 or c = F ยฐ, For any integer ~< let Dk(C)={(x,y)lx, y~C a,d d(x,y)=k}. The fami!y bk -I1/ICI)ID~(C)I, 0 <<-k <<-n, is called the distance distribu'ion of (2 If IC1~2 the number m(C)~min{k r bk 4=0} is called the minimum disi: ace of C. re(C) is a measure for the error-correcting capability of C. When F is a finite field, a code C ~ F" i:~ called linear it and only if C is a sub, pace of the vector space FL For any x~F" then w(x)=d(x 0) is the Hamming weight of x and a~ = [W~,(C)I, 0 ~< k < n, where W~(C) = {x [ :. ~-C and