Аннотация The Doob graph D(m, n), where m > 0, is the direct product of m copies of The Shrikhande graph and n copies of the complete graph K 4 on 4 vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). In this paper we consider MDS codes in Doob graphs with code distance d ≥ 3. We prove that if 2m + n > 6 and 2 < d < 2m + n, th...

At a seminar in the Bell Communications Research Colloquia Series, Dr. Richard W. Hamming, a Professor at the Naval Postgraduate School in Monterey, California and a retired Bell Labs scientist, gave a very interesting and stimulating talk, You and Your Research to an overflow audience of some 200 Bellcore staff members and visitors at the Morris Research and Engineering Center on March 7, 198...

In this paper we establish the connections between two di7erent extensions of Z4-linearity for binary Hamming spaces. We present both notions – propelinearity and G-linearity – in the context of isometries and group actions, taking the viewpoint of geometrically uniform codes extended to discrete spaces. We show a double inclusion relation: binary G-linear codes are propelinear codes, and trans...

We consider the symmetric group Sn as a metric space with the Hamming metric. The covering radius cr(S) of a set of permutations S ⊂ Sn is the smallest r such that Sn is covered by the balls of radius r centred at the elements of S. For given n and s, let f (n, s) denote the cardinality of the smallest set S of permutations with cr(S) 6 n − s. The value of f (n, 2) is the subject of a conjectur...

There are many nonisomorphic orthogonal arrays with parameters OA(s3, s2 + s+1, s, 2) although the existence of the arrays yields many restrictions. We denote this by OA(3, s) for simplicity. V.D. Tonchev showed that for even the case of s = 3, there are at least 68 nonisomorphic orthogonal arrays. The arrays that are constructed by the n−dimensional finite spaces have parameters OA(s, (s − 1)/...

Let F denote the finite field with two elements. We describe a construction of partitions of F , for n = 2 − 1, m ≥ 4, into cosets of pairwise distinct Hamming codes (we call such codes nonparallel) of length n. We give a lower bound for the number of different such partitions.

The q-round Rényi-Ulam pathological liar game with k lies on the set [n] := {1, . . . , n} is a 2-player perfect information zero sum game. In each round Paul chooses a subset A ⊆ [n] and Carole either assigns 1 lie to each element of A or to each element of [n]\A. Paul wins if after q rounds there is at least one element with k or fewer lies. The game is dual to the original Rényi-Ulam liar ga...

The Hamming distance between two permutations of a finite set X is the number of elements of X on which they differ. In the first part of this paper, we consider bounds for the cardinality of a subset (or subgroup) of a permutation group P on X with prescribed distances between its elements. In the second part. We consider similar results for sets of s-tuples of permutations; the role of Hammin...

Let d3(n, k) be the maximum possible minimum Hamming distance of a ternary [n, k, d; 3]-code for given values of n and k. It is proved that d3(44, 6) = 27, d3(76, 6) = 48, d3(94, 6) = 60, d3(124, 6) = 81, d3(130, 6) = 84, d3(134, 6) = 87, d3(138, 6) = 90, d3(148, 6) = 96, d3(152, 6) = 99, d3(156, 6) = 102, d3(164, 6) = 108, d3(170, 6) = 111, d3(179, 6) = 117, d3(188, 6) = 123, d3(206, 6) = 135,...

We give an explicit decoding scheme for the permutation arrays under Hamming distance metric, where the encoding is constructed via a distance-preserving mapping from ternary vectors to permutations (3-DPM).