Embedding in a perfect code

📝 Abstract

A binary 1-error-correcting code can always be embedded in a 1-perfect code of some larger length

💡 Analysis

A binary 1-error-correcting code can always be embedded in a 1-perfect code of some larger length

📄 Content

arXiv:0804.0006v3 [math.CO] 30 Apr 2016 Embedding in a perfect code⋆ Sergey V. Avgustinovich, Denis S. Krotov A binary 1-error-correcting code can always be embedded in a 1-perfect code of some larger length. For any 1-error-correcting binary code C of length m we will construct a 1-perfect binary code P(C) of length n = 2m −1 such that fixing the last n −m coordinates by zeroes in P(C) gives C. In particular, any complete or partial Steiner triple system (or any other system that forms a 1-code) can always be embedded in a 1-perfect code of some length (compare with [13]). Since the weight-3 words of a 1-perfect code P with 0n ∈P form a Steiner triple system, and the weight-4 words of an extended 1-perfect code P with 0n ∈P form a Steiner quadruple system, we have, as corollaries, the following well-known facts: a patrial Steiner triple (quadruple) system can always be embedded in a Steiner triple (quadruple) system [18] ([7]) (these results, as well as many other embedding theorems for Steiner systems, can be found in [10, 5]). Notation: • F m denotes the set of binary m-tuples, or binary m-words. • ˙F m := F m \ {0m}, where 0m is the all-zeroes m-word. • F m is considered as a vector space over GF(2) with calculations modulo 2. • Π = {π(1), . . . , π(m)} = {(10..0), . . . , (0..01)} is the natural basis in F m. • n := 2m −1. • The elements of F m will be denoted by Greek letters. • The elements of F n will be denoted by overlined letters, their coordinates being indexed by the elements of ˙F m, e.g., ¯w = {wι}ι∈˙F m; we assume that the first m coordinates have the indexes π(1), . . . , π(m), and the order of the other n −m indexes does not matter (but fixed). • {¯e(ι)}ι∈˙F m is the natural basis in F n; note that ¯e(π(ι)) = (π(ι), 0n−m). ⋆This is the peer reviewed version of the following article: “Embedding in a perfect code”, Journal of Combinatorial Designs 17(5) 2009, 419–423, which has been published in final form at http://dx.doi.org/10.1002/jcd.20207 . This article may be used for non-commercial purposes in ac- cordance with Wiley Terms and Conditions for Self-Archiving. The results of the paper were presented at the Workshop “Coding Theory Days in St. Petersburg”, Oct. 2008, St. Petersburg, Russia. The authors are with the Sobolev Institute of Mathematics, Novosibirsk, Russia. E-mail: {avgust,krotov}@math.nsc.ru The research was partially supported by the RFBR grant 07-01-00248 and 08-01-00673 1 • For any α = (α1, …, αm) ∈F m denote ¯α := (α, 0n−m); it also holds ¯α = Pm i=1 αi¯e(π(i)). • d(·, ·) denotes the Hamming distance between two words in F m or F n (the number of positions in which the words differ). • < . . . > denotes the linear span of the vectors or sets of vectors between the angle brackets. • The neighborhood Ω(M) of a set M ⊂F n is the set of vectors at distance at most 1 from M. • A set C ⊂F m is called a 1-code if the neighborhoods of the codewords are disjoint. • A 1-code P ⊂F n is called a 1-perfect code if Ω(P) = F n; in this case, |P| = 2n/(n+1). • The Hamming code H defined as H := {¯c ∈{0, 1}n| X α∈˙F m cαα = 0m} (1) is a linear 1-perfect code. • For any ι from ˙F m the linear ι-component of H is defined as Rι := {¯c ∈H|cα = cα+ι for all α ∈F m\ < ι >} (note that Rι is a linear subcode of H, for all ι). Since [19], linear components are used for constructing non-linear 1-perfect codes. For the first time, the method of synchronous switching nonintersecting linear i-components with different i, which is exploited in this paper (to follow our notations, we replace i by Greek letters), was used in [6]. This method was also used by different authors for constructing 1-perfect codes with specified properties, such as different ranks and(or) kernels [6, 14, 16, 3], trivial automorphism group [2, 11], nonsystematicity [1, 15, 12] (to read more about these and other results in the theory of 1-perfect codes, see [8, 17]). The term “switching” was introduced in [15]; probably, it came from the theory of Steiner triple systems. In general, this term refers to replacing some part of a 1-perfect code by some set with the same cardinality and same neighborhood; the resulting code will also be 1-perfect. With this approach, a large number of 1-perfect codes can be produced [9]. 1-Codes that admit replacing by other 1-codes with the same neighborhood deserve the independent study; they exist in F n for every odd n [20]. Being such a 1-code is the main property of Rι. Since our definition of linear components differs from others, we should prove this fact (in essence, the following lemma coincides with [6, Corollary 3.4]). Lemma 1. For any ¯z from F n it holds that Ω(Rι + ¯z) = Ω(Rι + ¯z + ¯e(ι)). 2 Proof: Without loss of generality, assume ¯z = 0n. Denote ¯e(0m) := 0n. Then, we have Ω(Rι) = [ κ∈F m (Rι + ¯e(κ)) = [ κ∈F m (Rι + ¯e(ι) + ¯e(κ+ι)) = [ λ∈F m ((Rι + ¯e(ι)) + ¯e(λ)) = Ω(Rι + ¯e(ι)) because ¯e(ι) + ¯e(κ) + ¯e(κ+ι) ∈Rι for all κ ∈F m. △ Lemma 2. Every element ¯c of < Rι, Rκ > satisfies cα + cα

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