Best Known (99−31, 99, s)-Nets in Base 32
(99−31, 99, 2186)-Net over F32 — Constructive and digital
Digital (68, 99, 2186)-net over F32, using
- 323 times duplication [i] based on digital (65, 96, 2186)-net over F32, using
- net defined by OOA [i] based on linear OOA(3296, 2186, F32, 31, 31) (dual of [(2186, 31), 67670, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3296, 32791, F32, 31) (dual of [32791, 32695, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3296, 32792, F32, 31) (dual of [32792, 32696, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3291, 32769, F32, 31) (dual of [32769, 32678, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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(325, 23, F32, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3296, 32792, F32, 31) (dual of [32792, 32696, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3296, 32791, F32, 31) (dual of [32791, 32695, 32]-code), using
- net defined by OOA [i] based on linear OOA(3296, 2186, F32, 31, 31) (dual of [(2186, 31), 67670, 32]-NRT-code), using
(99−31, 99, 4369)-Net in Base 32 — Constructive
(68, 99, 4369)-net in base 32, using
- 321 times duplication [i] based on (67, 98, 4369)-net in base 32, using
- net defined by OOA [i] based on OOA(3298, 4369, S32, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(3298, 65536, S32, 31), using
- discarding factors based on OA(3298, 65538, S32, 31), using
- discarding parts of the base [i] based on linear OA(25661, 65538, F256, 31) (dual of [65538, 65477, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [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(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- discarding parts of the base [i] based on linear OA(25661, 65538, F256, 31) (dual of [65538, 65477, 32]-code), using
- discarding factors based on OA(3298, 65538, S32, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(3298, 65536, S32, 31), using
- net defined by OOA [i] based on OOA(3298, 4369, S32, 31, 31), using
(99−31, 99, 36025)-Net over F32 — Digital
Digital (68, 99, 36025)-net over F32, using
(99−31, 99, large)-Net in Base 32 — Upper bound on s
There is no (68, 99, large)-net in base 32, because
- 29 times m-reduction [i] would yield (68, 70, large)-net in base 32, but