Article Contents
Article Contents

# Construction for both self-dual codes and LCD codes

• *Corresponding author: Keita Ishizuka
• From a given [n, k] code C, we give a method for constructing many [n, k] codes C' such that the hull dimensions of C and C' are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even [56, 28, 12] codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length 26 ≤ n ≤ 40.

Mathematics Subject Classification: Primary: 94B05; Secondary: 94B65.

 Citation:

• Figure 1.  Matrices $A_{37,22,5},A_{38,13,10}$

Table 1.  Inequivalent extremal doubly even $[56,28,12]$ codes

 y4 y8 y12 y16 y20 y24 total D11 45 19 15 2 33 4 118 C56,1 16 3 1 0 10 26 56 C56,2 34 27 26 1 3 14 105 C56,3 10 0 23 2 0 24 59 C56,4 10 109 17 58 2 16 212 C56,5 17 53 25 11 5 4 115

Table 2.  $P_i,S_i$ for $i = 1,2,3$

 i Pi Si 1 {2,5} {1,2,3,4,5,6,7,8,9,11} 2 {1,2,4} {3}

Table 3.  $C_{n,k,d}$ with $x,y$

 Cn,k,d x y C37,22,6 (010110011011111) (110010110000001) C38,13,11 (0010011100110001011000100) (1110100001110101110001101)

Table 4.  $d_{LCD}(n,k)$, where $26 \le n \le 40, 9 \le k \le 17$

 n\k 9 10 11 12 13 14 15 16 17 26 9 8 8 8 7 6 5-6 5 4 27 9 9 8 8 7 6 6 6 5 28 10 10 8 8 8 7 6 6 5-6 29 10 10 9 8 8 8 6 6 6 30 11 10 9-10 9 8 8 6-7 6 6 31 11 10 10 10 9 8 7-8 6-7 6 32 12 11 10 10 9-10 8-9 7-8 7-8 6-7 33 12 12 10-11 10 10 9-10 8-9 8 6-8 34 13 12 11-12 10-12 10 10 9-10 8-9 7-8 35 13-14 12-13 12 10-12 10-11 10 9-10 9-10 7-8 36 14 12-14 12-13 11-12 10-12 10-11 10 10 8-9 37 14 12-14 12-14 12-13 10-12 10-12 10-11 10 9-10 38 14-15 13-14 12-14 12-14 11*-12 10-12 10-12 10-11 9-10 39 14-16 14-15 13-14 12-14 11-13 11-12 10-12 10-12 10-11 40 15-16 14-16 13-15 13-14 12-14 11-13 10-12 10-12 10-12

Table 5.  dLCD(n, k), where 26 ≤ n ≤ 40, 18 ≤ k ≤ 26

 n\k 18 19 20 21 22 23 24 25 26 26 4 4 4 27 4 4 4 3 28 5 4 4 4 3 29 6 5 4 4 4 3 30 6 5 5 4 4 4 3 31 6 6 6 5 4 4 4 3 32 6 6 6 5-6 5 4 4 3-4 3 33 6-7 6 6 6 6 5 4 4 4 34 6-8 6-7 6 6 6 5-6 4 4 4 35 7-8 6-8 6-7 5-6 6 6 5 4 4 36 8 7-8 6-8 6-7 6 6 6 5 4 37 8-9 7-8 7-8 6-8 6*-7 6 6 5-6 5 38 9-10 8-9 8 7-8 6-8 6-7 6 6 6 39 10 9-10 8-9 7-8 7-8 6-8 6-7 6 6 40 10-11 9-10 9-10 7-9 8 7-8 6-8 6-7 6

Table 6.  dLCD(n, k), where 33 ≤ n ≤ 40, 27 ≤ k ≤ 34

 n\k 27 28 29 30 31 32 33 34 33 3 34 3-4 3 35 4 4 3 36 4 4 3-4 3 37 4 4 4 4 3 38 5 4 4 4 3-4 3 39 5-6 5 4 4 4 4 2-3 40 6 6 5 4 4 4 3-4 2-3
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