Best Known (125−30, 125, s)-Nets in Base 16
(125−30, 125, 8739)-Net over F16 — Constructive and digital
Digital (95, 125, 8739)-net over F16, using
- 1 times m-reduction [i] based on digital (95, 126, 8739)-net over F16, using
- net defined by OOA [i] based on linear OOA(16126, 8739, F16, 31, 31) (dual of [(8739, 31), 270783, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(16126, 131086, F16, 31) (dual of [131086, 130960, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16126, 131088, F16, 31) (dual of [131088, 130962, 32]-code), using
- trace code [i] based on linear OA(25663, 65544, F256, 31) (dual of [65544, 65481, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(25655, 65536, F256, 28) (dual of [65536, 65481, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- trace code [i] based on linear OA(25663, 65544, F256, 31) (dual of [65544, 65481, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16126, 131088, F16, 31) (dual of [131088, 130962, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(16126, 131086, F16, 31) (dual of [131086, 130960, 32]-code), using
- net defined by OOA [i] based on linear OOA(16126, 8739, F16, 31, 31) (dual of [(8739, 31), 270783, 32]-NRT-code), using
(125−30, 125, 131096)-Net over F16 — Digital
Digital (95, 125, 131096)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16125, 131096, F16, 30) (dual of [131096, 130971, 31]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16124, 131094, F16, 30) (dual of [131094, 130970, 31]-code), using
- trace code [i] based on linear OA(25662, 65547, F256, 30) (dual of [65547, 65485, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- trace code [i] based on linear OA(25662, 65547, F256, 30) (dual of [65547, 65485, 31]-code), using
- linear OA(16124, 131095, F16, 29) (dual of [131095, 130971, 30]-code), using Gilbert–Varšamov bound and bm = 16124 > Vbs−1(k−1) = 546 370313 171812 667769 093590 401328 782902 265080 887188 816463 016164 898903 488777 504794 065027 284076 403865 922800 777015 727430 955425 560634 217119 255704 027136 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(16124, 131094, F16, 30) (dual of [131094, 130970, 31]-code), using
- construction X with Varšamov bound [i] based on
(125−30, 125, large)-Net in Base 16 — Upper bound on s
There is no (95, 125, large)-net in base 16, because
- 28 times m-reduction [i] would yield (95, 97, large)-net in base 16, but