A new class of binary constant weight codes derived by groups of linear fractional mappings

Jun Imai, Yoshinao Shiraki

Research output: Contribution to journalArticle

Abstract

Let A(n, d, w) denote the maximum possible number of code words in binary (n,d,w) constant weight codes. For smaller instances of (n, d, w)s, many improvements have occurred over the decades. However, unknown instances still remain for larger (n, d, w)s (for example, those of n > 30 and d > 10). In this paper, we propose a new class of binary constant weight codes that fill in the remaining blank instances of (n, d, w)s. Specifically, we establish several new non-trivial lower bounds such as 336 for A(64, 12, 8), etc. (listed in Table 2). To obtain these results, we have developed a new systematic technique for construction by means of groups acting on some sets. The new technique is performed by considering a triad (G, Ω, f) ("Group G," "Set Ω," "Action f on Ω") simultaneously. Our results described in Sect. 3 are obtained by using permutations of the elements of a set that include ∞ homogeneously like the other elements, which play a role to improve their randomness. Specifically, in our examples, we adopt the following model such as (PGL2(q), 1(q), "linear fractional action of subgroups of PGL2(q) on 1(q)") as a typical construction model. Moreover, as an application, the essential examples in [7] constructed by using an alternating group are again reconstructed with our new technique of a triad model, after which they are all systematically understood in the context of finite subgroups that act fractionally on a projective space over a finite field.

Original languageEnglish
Pages (from-to)2481-2492
Number of pages12
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
VolumeE89-A
Issue number10
DOIs
Publication statusPublished - Oct 2006
Externally publishedYes

Fingerprint

Constant Weight Codes
Binary Code
Fractional
Subgroup
Alternating group
Projective Space
Randomness
Galois field
Table
Permutation
Model
Binary
Lower bound
Denote
Unknown
Class

Keywords

  • Binary constant weight codes
  • Linear fractional mappings
  • Lower bounds of the number of code words
  • Modular groups
  • Permutation representations
  • PGL(F )

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Hardware and Architecture
  • Information Systems

Cite this

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title = "A new class of binary constant weight codes derived by groups of linear fractional mappings",
abstract = "Let A(n, d, w) denote the maximum possible number of code words in binary (n,d,w) constant weight codes. For smaller instances of (n, d, w)s, many improvements have occurred over the decades. However, unknown instances still remain for larger (n, d, w)s (for example, those of n > 30 and d > 10). In this paper, we propose a new class of binary constant weight codes that fill in the remaining blank instances of (n, d, w)s. Specifically, we establish several new non-trivial lower bounds such as 336 for A(64, 12, 8), etc. (listed in Table 2). To obtain these results, we have developed a new systematic technique for construction by means of groups acting on some sets. The new technique is performed by considering a triad (G, Ω, f) ({"}Group G,{"} {"}Set Ω,{"} {"}Action f on Ω{"}) simultaneously. Our results described in Sect. 3 are obtained by using permutations of the elements of a set that include ∞ homogeneously like the other elements, which play a role to improve their randomness. Specifically, in our examples, we adopt the following model such as (PGL2(q), 1(q), {"}linear fractional action of subgroups of PGL2(q) on 1(q){"}) as a typical construction model. Moreover, as an application, the essential examples in [7] constructed by using an alternating group are again reconstructed with our new technique of a triad model, after which they are all systematically understood in the context of finite subgroups that act fractionally on a projective space over a finite field.",
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N2 - Let A(n, d, w) denote the maximum possible number of code words in binary (n,d,w) constant weight codes. For smaller instances of (n, d, w)s, many improvements have occurred over the decades. However, unknown instances still remain for larger (n, d, w)s (for example, those of n > 30 and d > 10). In this paper, we propose a new class of binary constant weight codes that fill in the remaining blank instances of (n, d, w)s. Specifically, we establish several new non-trivial lower bounds such as 336 for A(64, 12, 8), etc. (listed in Table 2). To obtain these results, we have developed a new systematic technique for construction by means of groups acting on some sets. The new technique is performed by considering a triad (G, Ω, f) ("Group G," "Set Ω," "Action f on Ω") simultaneously. Our results described in Sect. 3 are obtained by using permutations of the elements of a set that include ∞ homogeneously like the other elements, which play a role to improve their randomness. Specifically, in our examples, we adopt the following model such as (PGL2(q), 1(q), "linear fractional action of subgroups of PGL2(q) on 1(q)") as a typical construction model. Moreover, as an application, the essential examples in [7] constructed by using an alternating group are again reconstructed with our new technique of a triad model, after which they are all systematically understood in the context of finite subgroups that act fractionally on a projective space over a finite field.

AB - Let A(n, d, w) denote the maximum possible number of code words in binary (n,d,w) constant weight codes. For smaller instances of (n, d, w)s, many improvements have occurred over the decades. However, unknown instances still remain for larger (n, d, w)s (for example, those of n > 30 and d > 10). In this paper, we propose a new class of binary constant weight codes that fill in the remaining blank instances of (n, d, w)s. Specifically, we establish several new non-trivial lower bounds such as 336 for A(64, 12, 8), etc. (listed in Table 2). To obtain these results, we have developed a new systematic technique for construction by means of groups acting on some sets. The new technique is performed by considering a triad (G, Ω, f) ("Group G," "Set Ω," "Action f on Ω") simultaneously. Our results described in Sect. 3 are obtained by using permutations of the elements of a set that include ∞ homogeneously like the other elements, which play a role to improve their randomness. Specifically, in our examples, we adopt the following model such as (PGL2(q), 1(q), "linear fractional action of subgroups of PGL2(q) on 1(q)") as a typical construction model. Moreover, as an application, the essential examples in [7] constructed by using an alternating group are again reconstructed with our new technique of a triad model, after which they are all systematically understood in the context of finite subgroups that act fractionally on a projective space over a finite field.

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