Nonexistence of some Griesmer codes overFq

Non-existence of some 4-dimensional Griesmer codes over finite fields

“…Since (q − u)(q − 1) + q − r + 2 = i + u, the number of (q − r + 2)-lines through a fixed 0-point on the 0-line in δ is 1 + u/(r − 2 − u). So, p m divides u and r − 2 also from (12). From µ 0 = 1 and (11), we have…”

“…If h ≤ 2m, then, from (12) and (13), q divides either u or r − 1, a contradiction. Hence 2m ≤ h − 1.…”

“…We have recently proved the following. Theorem 1.1 ( [12]). There exists no [g q (4 As a continuation on the non-existence of Griesmer codes for k = 4, we prove the following four theorems.…”

On the Minimum Length of Linear Codes over the Field of 9 Elements

“…Setting i = 48, the maximum possible contributions of c j 's in (2.7) to the LHS of (4.10) are (c 47 , c 74 , c 76 , c 77 ) = (1, 5, 1, 2) for t = 0; (c 74 , c 76 , c 77 ) = (6, 1, 2) for t = 3 and (c 77 , c 78 ) = (1, 8) for t = 6, since a 78-plane has only 6-lines or 9-lines. Recall from Table 1 that the spectrum of a 48-plane is (τ 0 , τ 3 , τ 6 ) = (3,16,72 with j c j = 9. Suppose a 0 > 0.…”

“…For k = 3, n q (3, d) is known for all d for q 9. In this paper, we tackle the problem to determine n 9 (4, d) for all d. See [25] for the updated table of n q (k, d) for some small q and k. The following results are already known for n 9 (k, d) with k = 3, 4, see [5,14,15,16,17,19,21,24,25,27]. Theorem 1.5 ( [21,24]).…”