Perfect codes over graphs
نویسندگان
چکیده
منابع مشابه
Perfect codes in Doob graphs
We study 1-perfect codes in Doob graphsD(m,n). We show that such codes that are linear over GR(4) exist if and only if n = (4γ+δ−1)/3 andm = (4γ+2δ−4γ+δ)/6 for some integers γ ≥ 0 and δ > 0. We also prove necessary conditions on (m,n) for 1-perfect codes that are linear over Z4 (we call such codes additive) to exist in D(m,n) graphs; for some of these parameters, we show the existence of codes....
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A perfect code in a graph Γ = (V,E) is a subset C of V that is an independent set such that every vertex in V \ C is adjacent to exactly one vertex in C. A total perfect code in Γ is a subset C of V such that every vertex of V is adjacent to exactly one vertex in C. A perfect code in the Hamming graph H(n, q) agrees with a q-ary perfect 1-code of length n in the classical setting. A necessary a...
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We show that a 1-error-correcting code C is ‘foldable’ over its kernel via the Steiner triple systems associated to the codewords whenever C is perfect. The resulting ‘folding’ produces a graph invariant that for Vasil’ev codes of length 15 is complete, showing in particular that there exist nonadditive propelinear codes and just one nonlinear Vasil’ev additive code up to equivalence.
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An extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ with 9 ≥ κ ≥ 5 obtained via Solov’eva-Phelps doubling construction. Each of the 361 resulting graphs has most of its nonloop edges expressible in terms o...
متن کاملFull-Rank Perfect Codes over Finite Fields
In this paper, we propose a construction of fullrank q-ary 1-perfect codes over finite fields. This construction is a generalization of the Etzion and Vardy construction of fullrank binary 1-perfect codes (1994). Properties of i-components of q-ary Hamming codes are investigated and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 1986
ISSN: 0095-8956
DOI: 10.1016/0095-8956(86)90079-1