Best Known (55, 55+27, s)-Nets in Base 32
(55, 55+27, 2521)-Net over F32 — Constructive and digital
Digital (55, 82, 2521)-net over F32, using
- 322 times duplication [i] based on digital (53, 80, 2521)-net over F32, using
- net defined by OOA [i] based on linear OOA(3280, 2521, F32, 27, 27) (dual of [(2521, 27), 67987, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3280, 32774, F32, 27) (dual of [32774, 32694, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3280, 32776, F32, 27) (dual of [32776, 32696, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3279, 32769, F32, 27) (dual of [32769, 32690, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- 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(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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3280, 32776, F32, 27) (dual of [32776, 32696, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3280, 32774, F32, 27) (dual of [32774, 32694, 28]-code), using
- net defined by OOA [i] based on linear OOA(3280, 2521, F32, 27, 27) (dual of [(2521, 27), 67987, 28]-NRT-code), using
(55, 55+27, 24702)-Net over F32 — Digital
Digital (55, 82, 24702)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3282, 24702, F32, 27) (dual of [24702, 24620, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3282, 32784, F32, 27) (dual of [32784, 32702, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(3279, 32769, F32, 27) (dual of [32769, 32690, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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(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([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3282, 32784, F32, 27) (dual of [32784, 32702, 28]-code), using
(55, 55+27, large)-Net in Base 32 — Upper bound on s
There is no (55, 82, large)-net in base 32, because
- 25 times m-reduction [i] would yield (55, 57, large)-net in base 32, but