### 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 (PGL_{2}(_{q}), ^{1}(_{q}), "linear fractional action of subgroups of PGL_{2}(_{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 language | English |
---|---|

Pages (from-to) | 2481-2492 |

Number of pages | 12 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E89-A |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2006 |

Externally published | Yes |

### Fingerprint

### 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

**A new class of binary constant weight codes derived by groups of linear fractional mappings.** / Imai, Jun; Shiraki, Yoshinao.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E89-A, no. 10, pp. 2481-2492. https://doi.org/10.1093/ietfec/e89-a.10.2481

}

TY - JOUR

T1 - A new class of binary constant weight codes derived by groups of linear fractional mappings

AU - Imai, Jun

AU - Shiraki, Yoshinao

PY - 2006/10

Y1 - 2006/10

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.

KW - Binary constant weight codes

KW - Linear fractional mappings

KW - Lower bounds of the number of code words

KW - Modular groups

KW - Permutation representations

KW - PGL(F )

UR - http://www.scopus.com/inward/record.url?scp=33750074781&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750074781&partnerID=8YFLogxK

U2 - 10.1093/ietfec/e89-a.10.2481

DO - 10.1093/ietfec/e89-a.10.2481

M3 - Article

AN - SCOPUS:33750074781

VL - E89-A

SP - 2481

EP - 2492

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 10

ER -