Best Known (71−23, 71, s)-Nets in Base 32
(71−23, 71, 2980)-Net over F32 — Constructive and digital
Digital (48, 71, 2980)-net over F32, using
- 321 times duplication [i] based on digital (47, 70, 2980)-net over F32, using
- net defined by OOA [i] based on linear OOA(3270, 2980, F32, 23, 23) (dual of [(2980, 23), 68470, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3270, 32781, F32, 23) (dual of [32781, 32711, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3270, 32784, F32, 23) (dual of [32784, 32714, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- 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(3255, 32769, F32, 19) (dual of [32769, 32714, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,11]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3270, 32784, F32, 23) (dual of [32784, 32714, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3270, 32781, F32, 23) (dual of [32781, 32711, 24]-code), using
- net defined by OOA [i] based on linear OOA(3270, 2980, F32, 23, 23) (dual of [(2980, 23), 68470, 24]-NRT-code), using
(71−23, 71, 29117)-Net over F32 — Digital
Digital (48, 71, 29117)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3271, 29117, F32, 23) (dual of [29117, 29046, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3271, 32787, F32, 23) (dual of [32787, 32716, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(3267, 32768, F32, 23) (dual of [32768, 32701, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3252, 32768, F32, 18) (dual of [32768, 32716, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(324, 19, F32, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,32)), using
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- Reed–Solomon code RS(28,32) [i]
- discarding factors / shortening the dual code based on linear OA(324, 32, F32, 4) (dual of [32, 28, 5]-code or 32-arc in PG(3,32)), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(3271, 32787, F32, 23) (dual of [32787, 32716, 24]-code), using
(71−23, 71, large)-Net in Base 32 — Upper bound on s
There is no (48, 71, large)-net in base 32, because
- 21 times m-reduction [i] would yield (48, 50, large)-net in base 32, but