Best Known (128−31, 128, s)-Nets in Base 16
(128−31, 128, 8739)-Net over F16 — Constructive and digital
Digital (97, 128, 8739)-net over F16, using
- 162 times duplication [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
(128−31, 128, 131096)-Net over F16 — Digital
Digital (97, 128, 131096)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16128, 131096, F16, 31) (dual of [131096, 130968, 32]-code), using
- trace code [i] based on linear OA(25664, 65548, F256, 31) (dual of [65548, 65484, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [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 C([0,15]) ⊂ C([0,13]) [i] based on
- trace code [i] based on linear OA(25664, 65548, F256, 31) (dual of [65548, 65484, 32]-code), using
(128−31, 128, large)-Net in Base 16 — Upper bound on s
There is no (97, 128, large)-net in base 16, because
- 29 times m-reduction [i] would yield (97, 99, large)-net in base 16, but