45 research outputs found
Construction of Optimal Linear Codes by Geometric Puncturing
Dedicated to the memory of S.M. Dodunekov (1945–2012)Abstract. Geometric puncturing is a method to construct new codes. ACM Computing Classification System (1998): E.4.∗This research was partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 24540138
An Improvement to the Achievement of the Griesmer Bound
We denoted by nq(k, d), the smallest value of n for which an [n, k, d]q code exists for given q, k, d. Since nq(k, d) = gq(k, d) for all d ≥ dk + 1 for q ≥ k ≥ 3, it is a natural question whether the Griesmer bound is attained or not for d = dk , where gq(k, d) = ∑[d/q^i], i=0,...,k-1, dk = (k − 2)q^(k−1) − (k − 1)q^(k−2). It was shown by Dodunekov [2] and Maruta [9], [10] that there is no [gq(k, dk ), k, dk ]q code for q ≥ k, k = 3, 4, 5 and for q ≥ 2k − 3, k ≥ 6. The purpose of this paper is to determine nq(k, d) for d = dk as nq(k, d) = gq(k, d) + 1 for q ≥ k with 3 ≤ k ≤ 8 except for (k, q) = (7, 7), (8, 8), (8, 9).* This research was partially supported by Grant-in-Aid for Scientific Research of Japan
Society for the Promotion of Science under Contract Number 20540129
Note on an Improvement of the Griesmer Bound for q-ary Linear Codes
Let nq(k, d) denote the smallest value of n for which an [n, k, d]q code exists for given integers k and d with k ≥ 3, 1 ≤ d ≤ q^(k−1)
and a prime or a prime power q. The purpose of this note is to show that there exists a series of the functions h3,q, h4,q, ..., hk,q
such that nq(k, d) can be expressed.This research was partially supported by Grant-in-Aid for Scientific Research of Japan
Society for the Promotion of Science under Contract Number 20540129
Non–existence of some 4–dimensional Griesmer codes over finite fields
We prove the non--existence of codes for for , ; for ; for ; and with or for , where . This yields that for , and with for and that for for , and for , where denotes the minimum length for which an code exists
A Necessary and Sufficient Condition for the Existence of an (n,r)-arc in PG(2,q) and Its Applications
ACM Computing Classification System (1998): E.4.Let q be a prime or a prime power ≥ 3. The purpose of this
paper is to give a necessary and sufficient condition for the existence of
an (n, r)-arc in PG(2, q ) for given integers n, r and q using the geometric
structure of points and lines in PG(2, q ) for n > r ≥ 3. Using the geometric
method and a computer, it is shown that there exists no (34, 3) arc in
PG(2, 17), equivalently, there exists no [34, 3, 31] 17 code.This research was partially supported by Grant-in-Aid for Scientific Research of Japan
Society for the Promotion of Science under Contract Number 24540138