A New Class of Binary Constant Weight Codes Derived by Groups of Linear Fractional Mappings

Jun IMAI; Yoshinao SHIRAKI

doi:10.1093/ietfec/e89-a.10.2481

A New Class of Binary Constant Weight Codes Derived by Groups of Linear Fractional Mappings

Jun IMAI, Yoshinao SHIRAKI

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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(F_q), P¹(F_q), "linear fractional action of subgroups of PGL₂(F_q) on P¹(F_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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E89-A No.10 pp.2481-2492

Publication Date: 2006/10/01

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Online ISSN: 1745-1337

DOI: 10.1093/ietfec/e89-a.10.2481

Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)

Category: Coding Theory

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Jun IMAI, Yoshinao SHIRAKI, "A New Class of Binary Constant Weight Codes Derived by Groups of Linear Fractional Mappings" in IEICE TRANSACTIONS on Fundamentals, vol. E89-A, no. 10, pp. 2481-2492, October 2006, doi: 10.1093/ietfec/e89-a.10.2481.
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(F_q), P¹(F_q), "linear fractional action of subgroups of PGL₂(F_q) on P¹(F_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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e89-a.10.2481/_p

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@ARTICLE{e89-a_10_2481,
author={Jun IMAI, Yoshinao SHIRAKI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A New Class of Binary Constant Weight Codes Derived by Groups of Linear Fractional Mappings},
year={2006},
volume={E89-A},
number={10},
pages={2481-2492},
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(F_q), P¹(F_q), "linear fractional action of subgroups of PGL₂(F_q) on P¹(F_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.},
keywords={},
doi={10.1093/ietfec/e89-a.10.2481},
ISSN={1745-1337},
month={October},}

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TY - JOUR
TI - A New Class of Binary Constant Weight Codes Derived by Groups of Linear Fractional Mappings
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2481
EP - 2492
AU - Jun IMAI
AU - Yoshinao SHIRAKI
PY - 2006
DO - 10.1093/ietfec/e89-a.10.2481
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E89-A
IS - 10
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - October 2006
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(F_q), P¹(F_q), "linear fractional action of subgroups of PGL₂(F_q) on P¹(F_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.
ER -