Best Known (32−19, 32, s)-Nets in Base 128
(32−19, 32, 342)-Net over F128 — Constructive and digital
Digital (13, 32, 342)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (3, 22, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (1, 10, 150)-net over F128, using
(32−19, 32, 386)-Net over F128 — Digital
Digital (13, 32, 386)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12832, 386, F128, 19) (dual of [386, 354, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(12831, 382, F128, 19) (dual of [382, 351, 20]-code), using an extension Ce(18) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12828, 382, F128, 17) (dual of [382, 354, 18]-code), using an extension Ce(16) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1281, 4, F128, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
(32−19, 32, 514)-Net in Base 128 — Constructive
(13, 32, 514)-net in base 128, using
- base change [i] based on digital (9, 28, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- digital (0, 19, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 9, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(32−19, 32, 591714)-Net in Base 128 — Upper bound on s
There is no (13, 32, 591715)-net in base 128, because
- 1 times m-reduction [i] would yield (13, 31, 591715)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 210625 309036 785841 603665 425625 465454 081272 080897 228757 951192 273432 > 12831 [i]