Best Known (67, 67+27, s)-Nets in Base 32
(67, 67+27, 2565)-Net over F32 — Constructive and digital
Digital (67, 94, 2565)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- 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
- digital (1, 14, 44)-net over F32, using
(67, 67+27, 5042)-Net in Base 32 — Constructive
(67, 94, 5042)-net in base 32, using
- 323 times duplication [i] based on (64, 91, 5042)-net in base 32, using
- base change [i] based on (38, 65, 5042)-net in base 128, using
- 1281 times duplication [i] based on (37, 64, 5042)-net in base 128, using
- base change [i] based on digital (29, 56, 5042)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
- base change [i] based on digital (29, 56, 5042)-net over F256, using
- 1281 times duplication [i] based on (37, 64, 5042)-net in base 128, using
- base change [i] based on (38, 65, 5042)-net in base 128, using
(67, 67+27, 94121)-Net over F32 — Digital
Digital (67, 94, 94121)-net over F32, using
(67, 67+27, large)-Net in Base 32 — Upper bound on s
There is no (67, 94, large)-net in base 32, because
- 25 times m-reduction [i] would yield (67, 69, large)-net in base 32, but