Victor A. Zinoviev

Spherical codes generated by binary partitions of symmetric pointsets

Several constructions are presented by which spherical codes are generated from groups of binary codes. In the main family of constructions the codes are generated from equally spaced symmetric pointsets on the real line. The main ideas are code concatenation and set partitioning. Extensive tables are presented for spherical codes in dimension n/spl les/24. >

On Z_4 -Linear Goethals Codes and KloostermanSums

Studying the coset weight distributions of the Z4-linear Goethals codes, e connect these codes with the Kloosterman sums. From one side, e obtain for some cases, of the cosets of weight four, the exact expressions for the number of code ords of weight four in terms of the Kloosterman sums. From the other side, e obtain some limitations for the possible values of the Kloosterman sums, hich improve the well known results due to Lachaud and Wolfmann kn:lac.

On /spl Zopf//sub 4/-linear Preparata-like and Kerdock-like codes

We say that a binary code of length n is additive if it is isomorphic to a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /spl alpha/ + 2/spl beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /spl alpha/ = 0, i.e., it is always a /spl Zopf//sub 4/-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the /spl Zopf//sub 4/-dual of these codes, i.e., the /spl Zopf//sub 4/-linear Kerdock-like codes.

On non-antipodal binary completely regular codes

Binary non-antipodal completely regular codes are characterized. Using a result on nonexistence of nontrivial binary perfect codes, it is concluded that there are no unknown nontrivial non-antipodal completely regular binary codes with minimum distance d>=3. The only such codes are halves and punctured halves of known binary perfect codes. Thus, new such codes with covering radius @r=6 and 7 are obtained. In particular, a half of the binary Golay [23,12,7]-code is a new binary completely regular code with minimum distance d=8 and covering radius @r=7. The punctured half of the Golay code is a new completely regular code with minimum distance d=7 and covering radius @r=6. The new code with d=8 disproves the known conjecture of Neumaier, that the extended binary Golay [24,12,8]-code is the only binary completely regular code with d>=8. Halves of binary perfect codes with Hamming parameters also provide an infinite family of binary completely regular codes with d=4 and @r=3. Puncturing of these codes also provide an infinite family of binary completely regular codes with d=3 and @r=2. Both these families of codes are well known, since they are uniformly packed in the narrow sense, or extended such codes. Some of these completely regular codes are new completely transitive codes.

On Coset Weight Distributions of the 3-Error-Correcting BCH-Codes

We study the coset weight distributions of the 3-error-correcting binary narrow-sense BCH-codes and of their extensions, whose lengths are, respectively, 2m-1 and 2m, m odd. We prove that all weight distributions are known as soon as those of the cosets of minimum weight 4 of the extended code are known. We point out that properties of the cosets which are orphans yield interesting properties on the other cosets. We describe the classes of cosets which are equivalent under the affine permutations. At the end we produce significant numerical results, proving that the number of distinct weight distributions of cosets increases with the length of the codes.

On new completely regular q-ary codes

In this paper, new completely regular q-ary codes are constructed from q-ary perfect codes. In particular, several new ternary completely regular codes are obtained from the ternary [11, 6, 5] Golay code. One of these codes with parameters [11, 5, 6] has covering radius ρ = 5 and intersection array (22, 20, 18, 2, 1; 1, 2, 9, 20, 22). This code is dual to the ternary perfect [11, 6, 5] Golay code. Another [10, 5, 5] code has covering radius ρ = 4 and intersection array (20, 18, 4, 1; 1, 2, 18, 20). This code is obtained by deleting one position of the former code. All together, the ternary Golay code results in eight completely regular codes, only four of which were previously known. Also, new infinite families of completely regular codes are constructed from q-ary Hamming codes.

New Completely Regular

For any integer ¿ ¿ 1 and for any prime power q , an explicit construction of an infinite family of completely regular (and completely transitive) q-ary codes with d = 3 and with covering radius ¿ is given. The intersection array is also computed. Under the same conditions, the explicit construction of an infinite family of q-ary uniformly packed codes (in the wide sense) with covering radius ¿, which are not completely regular, is also given. In both constructions, the Kronecker product is the basic tool that has been used.

q

We characterize all

-ary Codes Based on Kronecker Products

q

On

-ary linear completely regular codes with covering radius