# BCH and Goppa-type binary linear sum-rank-metric codes This page lists some tables that provide the dimensions of binary linear sum-rank-metric codes constructed from two BCH codes and two Goppa codes over GF(4), respectively. For the constructions of SRM codes, we refer to the following references. 1. U. MartÍnez-Peňas, Sum-rank BCH codes and cyclic-skew-cyclic codes, IEEE Transactions on Information Theory, vol. 67, no. 8, pp. 51495167, 2021. 2. H. Chen, New explicit linear sum-rank-metric codes, IEEE Transactions on Information Theory, vol. 69, no. 10, pp. 6303-6313, 2023. 3. H. Chen, Z. Cheng and Y. Qi, Construction and fast decoding of binary linear sum-rank-metric codes, arXiv:2311.03619, 2023. #### 1. The dimensions of BCH-type binary linear sum-rank-metric codes with matrix size n\*m #### Table 1: Block length t=15, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Table IV)Singleton-like
42*242*202*27
52*212*182*26
62*202*162*25
72*172*142*24
82*152*102*23
92*132*82*22
102*132*82*21
112*112*62*20
122*102*42*19
132*82*22*18
142*82*22*17
152*62*22*16
#### Table 2： Block length t=31, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Table V)Singleton-like
42*552*502*59
52*472*472*58
62*462*452*57
72*412*402*56
82*402*352*55
92*322*322*54
102*322*302*53
112*312*272*52
122*302*252*51
132*222*222*50
142*222*202*49
152*212*152*48
182*122*102*45
222*122*72*41
262*72*22*37
302*72*22*33
#### Table 3： Block length t=63, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Table VI)Singleton-like
42*1182*1122*123
52*1132*1082*122
62*1122*1062*121
72*1062*1002*120
82*1032*942*119
92*992*902*118
102*982*882*117
112*942*822*116
122*912*762*115
132*862*722*114
142*852*702*113
152*792*642*112
162*762*582*111
172*722*542*110
182*722*542*109
192*712*542*108
202*712*542*107
212*682*542*106
222*672*522*105
232*632*502*104
242*602*442*103
252*562*402*102
262*562*402*101
272*542*382*100
282*512*322*99
292*462*282*98
302*462*282*97
312*422*262*96
322*392*202*95
382*352*162*89
462*292*82*81
542*202*22*73
#### Table 4： Block length t=127, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Table VII)Singleton-like
42*2452*2382*251
52*2332*2332*250
62*2322*2312*249
72*2252*2242*248
82*2242*2172*247
92*2122*2122*246
102*2112*2102*245
112*2042*2032*244
122*2032*1962*243
132*1912*1912*242
142*1902*1892*241
152*1832*1822*240
162*1822*1752*239
192*1692*1632*236
202*1682*1612*235
222*1552*1542*233
272*1342*1262*228
282*1332*1192*227
302*1202*1122*225
382*1002*912*217
402*992*842*215
462*792*702*209
542*652*442*201
602*502*352*195
622*442*282*193
762*302*92*179
802*302*92*175
922*232*92*163
1082*162*22*147
1202*92*22*135
1242*92*22*131
1262*92*22*129
#### Table 5： Block length t=255, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Theorem 9)Singleton-like
42*5002*4922*507
52*4932*4862*506
62*4922*4842*505
72*4842*4762*504
82*4802*4682*503
92*4742*4622*502
102*4732*4602*501
112*4682*4522*500
122*4642*4442*499
152*4482*4282*496
162*4442*4202*495
172*4382*4142*494
182*4372*4122*493
302*3862*3442*481
312*3782*3362*480
322*3742*3282*479
332*3692*3242*478
342*3692*3242*477
352*3662*3202*476
362*3642*3162*475
602*2682*1922*451
642*2482*1682*447
682*2442*1642*443
722*2392*1622*439
1002*1762*1122*411
1192*1312*682*392
1202*1292*662*391
1282*1032*382*383
1362*992*342*375
1442*972*322*367
1992*582*42*312
2002*582*42*311
2382*362*22*273
2392*342*22*272
2402*342*22*271
#### Table 6： Block length t=511, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Theorem 9)Singleton-like
42*10112*10022*1019
52*9952*9952*1018
62*9942*9932*1017
72*9852*9842*1016
82*9842*9752*1015
92*9682*9682*1014
102*9672*9662*1013
112*9582*9572*1012
122*9572*9482*1011
322*8222*8132*991
362*8042*7952*987
602*6512*6332*963
802*5672*5372*943
1002*4612*4382*923
1202*3782*3302*903
1402*3172*2692*883
1602*2952*2312*863
1802*2422*1862*843
2002*1882*1252*823
2202*1682*1052*803
2402*1292*632*783
2602*772*112*763
2802*772*112*743
3002*772*112*723
3202*772*112*703
3402*772*112*683
3602*592*112*663
3802*412*112*643
4002*322*22*623
4202*322*22*603
4402*312*22*583
4602*202*22*563
4802*192*22*543
5002*112*22*523
#### For decoding of BCH-type binary linear sum-rank-metric codes , the following binary linear sum-rank-metric codes with the block length 15, 63 ,255, and matrix size 2\*2, are constructed from quaternary BCH \[t, k\_i, d\_i] codes, satisfying d\_2>=2d\_1/3. #### Table 7： Block length t=15, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Table IV)Singleton-like
42*222*202*27
52*192*182*26
62*182*162*25
72*162*142*24
82*132*102*23
92*122*82*22
102*112*82*21
112*82*62*20
122*72*42*19
132*52*22*18
142*52*22*17
152*52*22*16
#### Table 8： Block length t=63, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Table VI)Singleton-like
42*1152*1122*123
52*1102*1082*122
62*1092*1062*121
72*1042*1002*120
82*1002*942*119
92*982*902*118
102*942*882*117
112*882*822*116
122*852*762*115
132*812*722*114
142*792*702*113
202*622*542*107
222*582*522*105
242*512*442*103
282*432*322*99
302*412*282*97
382*282*162*89
462*172*82*81
542*92*22*73
602*92*22*67
#### Table 9： Block length t=255, q=2, n=m=2
Minimum sum-rank distanceReference 3Reference 1 (Theorem 9)Singleton-like
42*4962*4922*507
52*4892*4862*506
62*4882*4842*505
72*4812*4762*504
82*4762*4682*503
92*4732*4622*502
102*4682*4602*501
112*4602*4522*500
122*4562*4442*499
152*4442*4282*496
162*4362*4202*495
172*4292*4142*494
182*4282*4122*493
302*3722*3442*481
312*3652*3362*480
322*3602*3282*479
332*3582*3242*478
342*3542*3242*477
352*3482*3202*476
362*3462*3162*475
602*2422*1922*451
642*2222*1682*447
682*2122*1642*443
722*2032*1622*439
1002*1382*1122*411
1192*1162*682*392
1202*1142*662*391
1282*992*382*383
1362*882*342*375
1442*752*322*367
1992*192*42*312
2002*192*42*311
2382*172*22*273
2392*172*22*272
2402*172*22*271
#### 2. The dimensions of Goppa type binary linear sum-rank-metric codes with matrix size n\*m #### Table 10： Block length t, q=2, n=m=2 | t | Minimum sum-rank distance | Reference 3 | Reference 1 (Tables) | Singleton-like | | --- | ------------------------- | ----------- | -------------------- | -------------- | | 32 | 5 | 2\*49 | t=31, dim=2\*47 | 2\*60 | | 32 | 18 | 2\*12 | t=31, dim=2\*7 | 2\*47 | | 32 | 22 | 2\*7 | t=31, dim=2\*7 | 2\*43 | | 32 | 26 | 2\*2 | t=31, dim=2\*2 | 2\*39 | | 64 | 5 | 2\*110 | t=63, dim=2\*108 | 2\*124 | | 128 | 5 | 2\*235 | t=127, dim=2\*233 | 2\*252 |