Abstract
In coding theory, the problem of finding the shortest linear codes for a fixed set of parameters is central. Given the dimension $k$, the minimum weight $d$, and the order $q$ of the finite field $\bF_q$ over which the code is defined, the function $n_q(k, d)$ specifies the smallest length $n$ for which an $[n, k, d]_q$ code exists. The problem of determining the values of this function is known as the problem of optimal linear codes. Using the geometric methods through projective geometry, we determine $n_q(4,d)$ for some values of $d$ by constructing new codes and by proving the nonexistence of linear codes with certain parameters.
Highlights
IntroductionWe denote by Fq the field of q elements. Let Fnq be the vector space of n-tuples over Fq
The problem of determining the values of this function is known as the problem of optimal linear codes
We denote by Fq the field of q elements
Summary
We denote by Fq the field of q elements. Let Fnq be the vector space of n-tuples over Fq. We conjecture that a [gq(4, d), 4, d]q code for d = 2q3 − rq2 − q + 1 with r = q − q/p − 1 does not exist for non-prime q ≥ 8, which is valid for q = 8, 9 by Theorem 1.5 and [17]. We need to show the existence of a [gq(4, d) + 1, 4, d]q code for d = q3 − 2q2 by Theorem 1.2 (b). This is already known for q = 3, 4, 5 and is valid for q = 8 by Theorem 1.5.
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