Let Fq be a finite field of q elements (q = pk, p is a primenumber). An infinite-dimensional q-ary vector space FN0q consists of allsequences u = (u1; u2; : : :), where ui 2 Fq and all ui are 0 except somefinite set of indices i 2N. A subset C FN0q is called a perfect q-arycode 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 H1q as the infinite unionof the sequence of finite q-ary codes eHnq where for all n = (qm??1)=(q??1),eHnq is a subcode of eHqn+1q . We prove that all linear perfect q-ary codesof infinite length are affine equivalent. A perfect q-ary code C FN0q iscalled systematic if N could be split into two subsets N1, N2 such that Cis a graphic of some function f : FN1;0q ! FN2;0q . Otherwise, C is callednonsystematic. Further general properties of systematic codes are proved.We also prove a version of Shapiro–Slotnik theorem for codes of infinitelength. Then, we construct nonsystematic codes of infinite length usingthe switchings of s q ?? 1 disjoint components. We say that a perfectcode C has the complete system of triples if for any three indices i1,i2, i3 the set C ?? C contains the vector with support fi1; i2; i3g. Weconstruct perfect codes of infinite length having the complete system oftriples (in particular, such codes are nonsystematic). These codes canbe obtained from the Hamming code H1q by switching some familyof disjoint components B = fRu11 ;Ru22 ; : : :g. 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 noncomplete system of triples.
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