Search

Abstract: All 708 103 nonisomorphic Steiner systems S(16, 4, 3) of order 16 and rank 14 over F 2 are enumerated. Among them there are exactly 1059 homogeneous systems. It is shown that all the 708 103 Steiner systems can be obtained by the general doubling construction presented in the paper.

All extended binary perfect (16, 4, 211) codes of rank 14 over the field F2are classified. It is proved that among all nonequivalent extended binary perfect (16, 4, 211) codes there are exactly 1719 nonequivalent codes of rank 14 over F2. Among these codes there are 844 codes classified by Phelps (Solov’eva-Phelps codes) and 875 other codes obtained by the construction of Etzion-Vardy and by a new general doubling construction, presented in the paper. Thus, the only open question in the classification of extended binary perfect (16,4,211) codes is that on such codes of rank 15 over F2.All extended binary perfect (16, 4, 211) codes of rank 14 over the field F2are classified. It is proved that among all nonequivalent extended binary perfect (16, 4, 211) codes there are exactly 1719 nonequivalent codes of rank 14 over F2. Among these codes there are 844 codes classified by Phelps (Solov’eva-Phelps codes) and 875 other codes obtained by the construction of Etzion-Vardy and by a new general doubling construction, presented in the paper. Thus, the only open question in the classification of extended binary perfect (16,4,211) codes is that on such codes of rank 15 over F2.

Extended binary perfect nonlinear Vasil’ev codes of length n= 2mand Steiner systems S(n, 4, 3) of rank n-mover F2are studied. The generalized concatenated construction of Vasil’ev codes induces a variant of the doubling construction for Steiner systems S(n, 4, 3) of an arbitrary rank rover F2. We prove that any Steiner system S(n= 2m, 4, 3) of rank n-mcan be obtained by this doubling construction and is formed by codewords of weight 4 of these Vasil’ev codes. The length 16 is studied in detail. Orders of the full automorphism groups of all 12 nonequivalent Vasil’ev codes of length 16 are found. There are exactly 15 nonisomorphic systems S(16, 4, 3) of rank 12 over F2, and they can be obtained from codewords of weight 4 of the extended Vasil’ev codes. Orders of the automorphism groups of all these Steiner systems are found.

A Steiner quadruple system SQS(v) of order vis a 3-design T(v, 4, 3, λ) with λ = 1. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over $$ \mathbb{F} $$2. In particular, there is one system SQS(16) of rank 11 (points and planes of the a fine geometry AG(4, 2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

A Steiner quadruple system SQS(v) of order v is a 3-design T (v; 4; 3;λ) with λ = 1. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over F2. In particular, there is one system SQS(16) of rank 11 (points and planes of the a fine geometry AG(4; 2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

Composing the Carlet map with the inverse Gray map, a new family of cyclic quaternary codes is constructed from 5-cyclic $$\mathbb{Z}_8 $$ -codes. This new family of codes is inspired by the quaternary Shanbag–Kumar–Helleseth family, a recent improvement on the Delsarte–Goethals family. We conjecture that these $$\mathbb{Z}_4 $$ -codes are not linear. As applications, we construct families of low-correlation quadriphase and biphase sequences.

We enumerate binary nonlinear perfect codes of length 15 obtained by the generalized concatenated (GC) construction. There are 15 different types of such codes. They are defined by pairs of MDS codes Ai: (4, 2, 64)4. For every pair we give the number of nonequivalent codes of this type. In total, there are 777 nonequivalent binary perfect codes of length 15 obtained by the GC construction. This number includes the Hamming code (of rank 11), 18 Vasil'ev codes (of rank 12), and 758 codes of rank 13.

We enumerate binary extended nonlinear perfect codes of length 16 obtained by the generalized concatenated construction (GC-construction). There are 15 different types of such codes. They are defined by pairs of MDS codes Ai: (4, 2, 64)4. For every pair, we give the number of nonequivalent codes of this type. In total, there are 285 nonequivalent binary extended nonlinear perfect codes of length 16 obtained by the GC-construction, including the Hamming (i.e., linear) code. Thus, we obtain all binary extended perfect codes of length 16 and rank 13. Their total number is equal to 272.