On the Minimum Length of some Linear Codes of Dimension 5

Abstract

In this paper, we shall prove that the minimum length n q (5, d) is equal to g q (5, d)

On the minimum length of linear codes overF5

“…It is known that the spectrum of a [244, 4,195] 5 code is (a 24 , a 44 , a 49 ) = (2, 6, 148) by Lemma 5.4 in [25] and that a [243, 4,194] 5 code is extendable by Lemma 10 in [7]. Actually, Lemma 10 in [7] says that the multiset for a [243, 4,194] 5 code is 2Σ − (δ 1 + δ 2 − ℓ − Q ), where Σ = PG(3, 5); δ 1 and δ 2 are distinct planes; and ℓ and Q are a line and a point, respectively, such that ℓ ∩ δ 1 ∩ δ 2 = ∅ and Q ̸ ∈ (δ 1 ∩ δ 2 ) ∪ (ℓ ∩ δ 1 ) ∪ (ℓ ∩ δ 2 ).…”

“…Actually, Lemma 10 in [7] says that the multiset for a [243, 4,194] 5 code is 2Σ − (δ 1 + δ 2 − ℓ − Q ), where Σ = PG(3, 5); δ 1 and δ 2 are distinct planes; and ℓ and Q are a line and a point, respectively, such that ℓ ∩ δ 1 ∩ δ 2 = ∅ and Q ̸ ∈ (δ 1 ∩ δ 2 ) ∪ (ℓ ∩ δ 1 ) ∪ (ℓ ∩ δ 2 ). Lemma 4.20.…”

“…In this paper, we tackle the problem to determine n 5 (5, d) for all d. See [24] for the updated table of n 5 (5, d). The following results are already known for n 5 (k, d), see [1][2][3]5,7,9,11,21,20,25,24,30]. (3) n 5 (4, d) = g 5 (4, d) + 2 for d = 25.…”

On the minimum length of some linear codes

“…So, (1) and (3) of Theorem A are not valid for q = 2. However, (2) of Theorem A holds for k = 6, 8 (but not for k = 4) for q = 2 [1].…”