p | nbr of Euclidean LCD skew cyc. | nbr of Hermitian LCD skew cyc. | ||
3 | 18 | 16 | 36 | 32 |
5 | 3750 | 2412 | 3750 | 2412 |
7 | 705984 | 39564 | 941192 | 52752 |
11 | 259374246010 | 311249095212 |
This paper is about a first characterization of LCD skew constacyclic codes and some constructions of LCD skew cyclic and skew negacyclic codes over $ \mathbb{F}_{p^2} $.
Citation: |
Table 1.
Number of Euclidean and Hermitian LCD
p | nbr of Euclidean LCD skew cyc. | nbr of Hermitian LCD skew cyc. | ||
3 | 18 | 16 | 36 | 32 |
5 | 3750 | 2412 | 3750 | 2412 |
7 | 705984 | 39564 | 941192 | 52752 |
11 | 259374246010 | 311249095212 |
Table 5.
Dimensions of MDS LCD skew codes over
MDS Euclidean LCD | MDS Hermitian LCD | |||
length | skew cyc | skew nega | skew cyc | skew nega |
4 | 2 | 2 | no | 2 |
6 | 3 | 3 | 3 | no |
8 | 3, 4, 5 | 4 | 3, 5 | 4 |
10 | 5 | 5 | 5 | no |
Table 6.
Dimensions of MDS LCD skew codes over
MDS Euclidean LCD | MDS Hermitian LCD | |||
length | skew cyc | skew nega | skew cyc | skew nega |
4 | 2 | 2 | no | 2 |
6 | 2, 3, 4 | 2, 3, 4 | 2, 3, 4 | 2, 4 |
8 | 3, 4, 5 | 4 | 3, 5 | 4 |
10 | 5 | 5 | 5 | no |
12 | 3, 5, 6, 7, 9 | 6 | 3, 5, 7, 9 | 6 |
14 | 7 | 7 | 7 | no |
16 | 7, 8, 9 | no | 7, 9 | no |
18 | 9 | 9 | 9 | no |
Table 7.
Dimensions of MDS LCD skew codes over
MDS Euclidean LCD | MDS Hermitian LCD | |||
length | skew cyc | skew nega | skew cyc | skew nega |
4 | 2 | 2 | no | 2 |
6 | 2, 3, 4 | 2, 3, 4 | 2, 3, 4 | 2, 4 |
8 | 3, 4, 5 | 2, 4, 6 | 3, 5 | 2, 4, 6 |
10 | 4, 5, 6 | 4, 5, 6 | 4, 5, 6 | 4, 6 |
12 | 3, 5, 6, 7, 9 | 6 | 3, 5, 7, 9 | 6 |
14 | 7 | 7 | 7 | no |
16 | 3, 5, 7, 8, 9, 11, 13 | 8 | 3, 5, 7, 9, 11, 13 | 8 |
Table 2.
Best minimum distances and numbers of Euclidean LCD
Euclidean LCD skew cyc. | Euclidean LCD skew cyc. | ||||
length | best dist | nbr | length | best dist | nbr |
2 | 2* | 2 | 26 | 9 | 8 064 |
4 | 3* | 4 | 28 | 11* | 18 432 |
6 | 4* | 4 | 30 | 12* | 13 056 |
8 | 4* | 16 | 32 | 10 | 65 536 |
10 | 5* | 24 | 34 | 11* | 115 200 |
12 | 5 | 32 | 36 | 11 | 114 688 |
14 | 6* | 144 | 38 | 12* | 523 264 |
16 | 6 | 256 | 40 | 12* | 786 432 |
18 | 7 | 224 | 42 | 12 | 1 198 080 |
20 | 8* | 768 | 44 | 13 | 4 063 232 |
22 | 8* | 1 984 | 46 | 14* | 8 392 704 |
24 | 9* | 2 048 | 48 | 14* | 8 388 608 |
Table 3.
Best minimum distances and numbers of LCD
Euclidean LCD | ||||
skew cyc | skew negacyc | |||
length | best dist | nbr | best dist | nbr |
2 | 2* | 2 | 2* | 4 |
4 | 3* | 32 | 3* | 6 |
6 | 4* | 18 | 4* | 36 |
8 | 5* | 192 | 5* | 90 |
10 | 6* | 144 | 6* | 288 |
12 | 6* | 5 408 | 6* | 486 |
14 | 7* | 1 404 | 7* | 2 808 |
16 | 7 | 17 280 | 8* | 6 642 |
18 | 9* | 13 122 | 9* | 26 244 |
20 | 9 | 165 888 | 9 | 39 852 |
22 | 9* | 118 584 | 9* | 237 168 |
24 | 10* | 2 628 288 | 10* | 590 490 |
Table 4.
Best minimum distances and numbers of LCD
Hermitian LCD | ||||
skew cyc | skew negacyc | |||
length | best dist | nbr | best dist | nbr |
2 | 2* | 4 | 0 | 0 |
4 | 0 | 0 | 3* | 6 |
6 | 4* | 361 | 0 | 0 |
8 | 0 | 0 | 5* | 90 |
10 | 6* | 288 | 0 | 0 |
12 | 0 | 0 | 6* | 486 |
14 | 7* | 2 808 | 0 | 0 |
16 | 0 | 0 | 8* | 6 642 |
18 | 9* | 26 244 | 0 | 0 |
20 | 0 | 0 | 10* | 39 852 |
22 | 9* | 237 168 | 0 | 0 |
24 | 0 | 0 | 10* | 590 490 |
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