Abstract: Let Fq be a finite field of q elements (q = pk, p is a prime number). An infinite-dimensional q-ary vector space consists of all sequences u = (u1, u2, .), where ui ? Fq and all ui are 0 except some finite set of indices i ? N. A subset is called a perfect q-ary code with distance 3 if all balls of radius 1 (in the Hamming metric) with centers in C are pairwise disjoint and their union covers the space. Define the infinite perfect q-ary Hamming code as the infinite union of the sequence of finite q-axy codes where for all n = (qm-1)/(q-1), is a subcode of. We prove that all linear perfect q-ary codes of infinite length are affine equivalent. A perfect q-ary code is called systematic if N could be split into two subsets N1, N2 such that C is a graphic of some function. Otherwise, C is called nonsystematic. Further general properties of systematic codes are proved. Wc also prove a version of Shapiro Slotnik theorem far codes of infinite length. Then, we construct nonsystematic codes of infinite length using the switchings of s < q-1 disjoint components. We say that a perfect code C has the complete system of triples if for any three indices i1, i2, i3 the set C C contains the vector with support [i1, i2, i3]-construct perfect codes of infinite length having the complete system of triples (in particular, such codes are nonsystematic). These codes can be obtained from the Hamming code by switching some family of disjoint components. Unlike the codes of finite length, the family B must obey the rigid condition of sparsity. It is shown particularly that if the family of components B does not satisfy the condition of sparsity then it can generate a perfect code having non-complete system of triples. © 2019.

Cite: Malyugin S.A.
Systematic and nonsystematic perfect codes of infinite length over finite fields
Сибирские электронные математические известия (Siberian Electronic Mathematical Reports). 2019. V.16. P.1732-1751. DOI: 10.33048/semi.2019.16.122 WOS Scopus OpenAlex

Identifiers:

Web of science:	WOS:000501163400002
Scopus:	2-s2.0-85090407956
OpenAlex:	W3016243433

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Scopus	3
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