LCD and non-LCD (A)MDS DTRS codes with all possible hooks

Kapish Chand Meena; Ambrish Awasthi; Rajendra Kumar Sharma

doi:10.3934/amc.2026019

Article Contents

2026, Volume 23: 189-208. Doi: 10.3934/amc.2026019

This volume Previous Article Binary cyclic codes from three classes of sequences Next Article New infinite families of uniformly packed near-MDS codes and multiple coverings, based on the ternary Golay code

LCD and non-LCD (A)MDS DTRS codes with all possible hooks

1.
Scientific Analysis Group, Defence Research and Development Organisation, Delhi-110054, India
2.
Department of Applied Mathematics, Delhi Technological University, Delhi-110042, India

^*Corresponding author: Rajendra K. Sharma
^*Corresponding author: Rajendra K. Sharma

Received: May 2025

Revised: December 23, 2025

Early access: February 6, 2026

Published online: February 6, 2026

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Abstract

In this paper, we study $ k $-dimensional double-twisted Reed-Solomon (DTRS) codes over the finite fields $ \mathbb{F}_q $, with twists $ {\mathit{\boldsymbol{t}}} = (1, 2) $ and hooks $ {\mathit{\boldsymbol{h}}} = (h_0, h_1) $ where $ 0 \leq h_0 \leq h_{1}\leq k-1 $. We give necessary and sufficient conditions for such DTRS codes to be either MDS or AMDS. In addition, we provide a SageMath implementation of these conditions. We also give some examples of such DTRS codes along with their counting. Further, we show the existence of the hull of dimensions up to $ 3 $ for such codes. In particular, we give the existence of LCD and non-LCD DTRS codes for all possible hooks.

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Mathematics Subject Classification: Primary: 11T06, 11T71, 12F05, 94A24, 94B05.

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Table 1. Number of $ [n, k] $ DTRS codes satisfying Theorem 3.2

$ q $	$ n $	$ h_0 $	$ h_1 $	$ k =2 $	$ k=3 $	$ k=4 $	$ k=5 $	$ q $	$ n $	$ h_0 $	$ h_1 $	$ k =2 $	$ k=3 $	$ k=4 $	$ k=5 $
$ 5 $	$ 4 $	$ 0 $	$ 0 $	$ 72 $				$ 8 $	$ 4 $	$ 0 $	$ 1 $	$ 3325 $
$ 5 $	$ 4 $	$ 0 $	$ 1 $	$ 62 $				$ 8 $	$ 5 $	$ 0 $	$ 1 $	$ 2534 $	$ 2464 $
$ 5 $	$ 4 $	$ 1 $	$ 1 $	$ 64 $				$ 8 $	$ 6 $	$ 0 $	$ 1 $	$ 1162 $	$ 952 $	$ 1162 $
$ 7 $	$ 4 $	$ 0 $	$ 0 $	$ 1188 $				$ 8 $	$ 7 $	$ 0 $	$ 1 $	$ 287 $	$ 133 $	$ 182 $	$ 378 $
$ 7 $	$ 5 $	$ 0 $	$ 0 $	$ 648 $	$ 720 $			$ 8 $	$ 5 $	$ 0 $	$ 2 $		$ 2506 $
$ 7 $	$ 6 $	$ 0 $	$ 0 $	$ 180 $	$ 216 $	$ 240 $		$ 8 $	$ 6 $	$ 0 $	$ 2 $		$ 1050 $	$ 1162 $
$ 7 $	$ 4 $	$ 0 $	$ 1 $	$ 1146 $				$ 8 $	$ 7 $	$ 0 $	$ 2 $		$ 189 $	$ 203 $	$ 378 $
$ 7 $	$ 5 $	$ 0 $	$ 1 $	$ 600 $	$ 696 $			$ 8 $	$ 6 $	$ 0 $	$ 3 $			$ 1204 $
$ 7 $	$ 6 $	$ 0 $	$ 1 $	$ 156 $	$ 200 $	$ 213 $		$ 8 $	$ 7 $	$ 0 $	$ 3 $			$ 238 $	$ 378 $
$ 7 $	$ 5 $	$ 0 $	$ 2 $		$ 738 $			$ 8 $	$ 7 $	$ 0 $	$ 4 $				$ 378 $
$ 7 $	$ 6 $	$ 0 $	$ 2 $		$ 234 $	$ 240 $		$ 8 $	$ 4 $	$ 1 $	$ 1 $	$ 3220 $
$ 7 $	$ 6 $	$ 0 $	$ 3 $			$ 213 $		$ 8 $	$ 5 $	$ 1 $	$ 1 $	$ 2324 $	$ 2548 $
$ 7 $	$ 4 $	$ 1 $	$ 1 $	$ 1140 $				$ 8 $	$ 6 $	$ 1 $	$ 1 $	$ 952 $	$ 1078 $	$ 1225 $
$ 7 $	$ 5 $	$ 1 $	$ 1 $	$ 576 $	$ 666 $			$ 8 $	$ 7 $	$ 1 $	$ 1 $	$ 182 $	$ 196 $	$ 266 $	$ 336 $
$ 7 $	$ 6 $	$ 1 $	$ 1 $	$ 138 $	$ 168 $	$ 216 $		$ 8 $	$ 5 $	$ 1 $	$ 2 $		$ 2576 $
$ 7 $	$ 5 $	$ 1 $	$ 2 $		$ 648 $			$ 8 $	$ 6 $	$ 1 $	$ 2 $		$ 1141 $	$ 1246 $
$ 7 $	$ 6 $	$ 1 $	$ 2 $		$ 159 $	$ 232 $		$ 8 $	$ 7 $	$ 1 $	$ 2 $		$ 266 $	$ 287 $	$ 336 $
$ 7 $	$ 6 $	$ 1 $	$ 3 $			$ 228 $		$ 8 $	$ 6 $	$ 1 $	$ 3 $			$ 1267 $
$ 7 $	$ 5 $	$ 2 $	$ 2 $		$ 684 $			$ 8 $	$ 7 $	$ 1 $	$ 3 $			$ 301 $	$ 336 $
$ 7 $	$ 6 $	$ 2 $	$ 2 $		$ 180 $	$ 216 $		$ 8 $	$ 7 $	$ 1 $	$ 4 $				$ 357 $
$ 7 $	$ 6 $	$ 2 $	$ 3 $			$ 201 $		$ 8 $	$ 5 $	$ 2 $	$ 2 $		$ 2548 $
$ 7 $	$ 6 $	$ 3 $	$ 3 $			$ 228 $		$ 8 $	$ 6 $	$ 2 $	$ 2 $		$ 1078 $	$ 1246 $
$ 8 $	$ 4 $	$ 0 $	$ 0 $	$ 3290 $				$ 8 $	$ 7 $	$ 2 $	$ 2 $		$ 196 $	$ 280 $	$ 336 $
$ 8 $	$ 5 $	$ 0 $	$ 0 $	$ 2464 $	$ 2632 $			$ 8 $	$ 6 $	$ 2 $	$ 3 $			$ 1246 $
$ 8 $	$ 6 $	$ 0 $	$ 0 $	$ 1092 $	$ 1204 $	$ 1309 $		$ 8 $	$ 7 $	$ 2 $	$ 3 $			$ 266 $	$ 336 $
$ 8 $	$ 7 $	$ 0 $	$ 0 $	$ 252 $	$ 280 $	$ 329 $	$ 378 $	$ 8 $	$ 7 $	$ 2 $	$ 4 $				$ 336 $
$ 8 $	$ 6 $	$ 3 $	$ 3 $			$ 1225 $		$ 8 $	$ 7 $	$ 3 $	$ 4 $				$ 336 $
$ 8 $	$ 7 $	$ 3 $	$ 3 $			$ 245 $	$ 336 $	$ 8 $	$ 7 $	$ 4 $	$ 4 $				$ 336 $

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