Best Known (88−21, 88, s)-Nets in Base 16
(88−21, 88, 13109)-Net over F16 — Constructive and digital
Digital (67, 88, 13109)-net over F16, using
- net defined by OOA [i] based on linear OOA(1688, 13109, F16, 21, 21) (dual of [(13109, 21), 275201, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1688, 131091, F16, 21) (dual of [131091, 131003, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1688, 131096, F16, 21) (dual of [131096, 131008, 22]-code), using
- trace code [i] based on linear OA(25644, 65548, F256, 21) (dual of [65548, 65504, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,10]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(25644, 65548, F256, 21) (dual of [65548, 65504, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1688, 131096, F16, 21) (dual of [131096, 131008, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1688, 131091, F16, 21) (dual of [131091, 131003, 22]-code), using
(88−21, 88, 131096)-Net over F16 — Digital
Digital (67, 88, 131096)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1688, 131096, F16, 21) (dual of [131096, 131008, 22]-code), using
- trace code [i] based on linear OA(25644, 65548, F256, 21) (dual of [65548, 65504, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,10]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(25644, 65548, F256, 21) (dual of [65548, 65504, 22]-code), using
(88−21, 88, large)-Net in Base 16 — Upper bound on s
There is no (67, 88, large)-net in base 16, because
- 19 times m-reduction [i] would yield (67, 69, large)-net in base 16, but