Best Known (56−13, 56, s)-Nets in Base 32
(56−13, 56, 174807)-Net over F32 — Constructive and digital
Digital (43, 56, 174807)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (36, 49, 174763)-net over F32, using
- net defined by OOA [i] based on linear OOA(3249, 174763, F32, 13, 13) (dual of [(174763, 13), 2271870, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3249, 1048579, F32, 13) (dual of [1048579, 1048530, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 1048580, F32, 13) (dual of [1048580, 1048531, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(3249, 1048576, F32, 13) (dual of [1048576, 1048527, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 1048580, F32, 13) (dual of [1048580, 1048531, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3249, 1048579, F32, 13) (dual of [1048579, 1048530, 14]-code), using
- net defined by OOA [i] based on linear OOA(3249, 174763, F32, 13, 13) (dual of [(174763, 13), 2271870, 14]-NRT-code), using
- digital (1, 7, 44)-net over F32, using
(56−13, 56, 349527)-Net in Base 32 — Constructive
(43, 56, 349527)-net in base 32, using
- base change [i] based on digital (27, 40, 349527)-net over F128, using
- 1281 times duplication [i] based on digital (26, 39, 349527)-net over F128, using
- net defined by OOA [i] based on linear OOA(12839, 349527, F128, 13, 13) (dual of [(349527, 13), 4543812, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(12839, 2097163, F128, 13) (dual of [2097163, 2097124, 14]-code), using
- net defined by OOA [i] based on linear OOA(12839, 349527, F128, 13, 13) (dual of [(349527, 13), 4543812, 14]-NRT-code), using
- 1281 times duplication [i] based on digital (26, 39, 349527)-net over F128, using
(56−13, 56, 1803161)-Net over F32 — Digital
Digital (43, 56, 1803161)-net over F32, using
(56−13, 56, large)-Net in Base 32 — Upper bound on s
There is no (43, 56, large)-net in base 32, because
- 11 times m-reduction [i] would yield (43, 45, large)-net in base 32, but