Best Known (51, 76, s)-Nets in Base 32
(51, 76, 2731)-Net over F32 — Constructive and digital
Digital (51, 76, 2731)-net over F32, using
- 322 times duplication [i] based on digital (49, 74, 2731)-net over F32, using
- net defined by OOA [i] based on linear OOA(3274, 2731, F32, 25, 25) (dual of [(2731, 25), 68201, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3274, 32773, F32, 25) (dual of [32773, 32699, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3274, 32776, F32, 25) (dual of [32776, 32702, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(3273, 32769, F32, 25) (dual of [32769, 32696, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3267, 32769, F32, 23) (dual of [32769, 32702, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3274, 32776, F32, 25) (dual of [32776, 32702, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3274, 32773, F32, 25) (dual of [32773, 32699, 26]-code), using
- net defined by OOA [i] based on linear OOA(3274, 2731, F32, 25, 25) (dual of [(2731, 25), 68201, 26]-NRT-code), using
(51, 76, 24603)-Net over F32 — Digital
Digital (51, 76, 24603)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3276, 24603, F32, 25) (dual of [24603, 24527, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3276, 32784, F32, 25) (dual of [32784, 32708, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3273, 32769, F32, 25) (dual of [32769, 32696, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3261, 32769, F32, 21) (dual of [32769, 32708, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3276, 32784, F32, 25) (dual of [32784, 32708, 26]-code), using
(51, 76, large)-Net in Base 32 — Upper bound on s
There is no (51, 76, large)-net in base 32, because
- 23 times m-reduction [i] would yield (51, 53, large)-net in base 32, but