Best Known (23−13, 23, s)-Nets in Base 128
(23−13, 23, 387)-Net over F128 — Constructive and digital
Digital (10, 23, 387)-net over F128, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (0, 6, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 13, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 4, 129)-net over F128, using
(23−13, 23, 465)-Net over F128 — Digital
Digital (10, 23, 465)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12823, 465, F128, 13) (dual of [465, 442, 14]-code), using
- 80 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 77 times 0) [i] based on linear OA(12821, 383, F128, 13) (dual of [383, 362, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12821, 382, F128, 13) (dual of [382, 361, 14]-code), using an extension Ce(12) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12820, 382, F128, 12) (dual of [382, 362, 13]-code), using an extension Ce(11) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1280, 1, F128, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- 80 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 77 times 0) [i] based on linear OA(12821, 383, F128, 13) (dual of [383, 362, 14]-code), using
(23−13, 23, 515)-Net in Base 128 — Constructive
(10, 23, 515)-net in base 128, using
- (u, u+v)-construction [i] based on
- (1, 7, 257)-net in base 128, using
- 1 times m-reduction [i] based on (1, 8, 257)-net in base 128, using
- base change [i] based on digital (0, 7, 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
- base change [i] based on digital (0, 7, 257)-net over F256, using
- 1 times m-reduction [i] based on (1, 8, 257)-net in base 128, using
- (3, 16, 258)-net in base 128, using
- base change [i] based on digital (1, 14, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 14, 258)-net over F256, using
- (1, 7, 257)-net in base 128, using
(23−13, 23, 546)-Net in Base 128
(10, 23, 546)-net in base 128, using
- (u, u+v)-construction [i] based on
- (1, 7, 257)-net in base 128, using
- 1 times m-reduction [i] based on (1, 8, 257)-net in base 128, using
- base change [i] based on digital (0, 7, 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
- base change [i] based on digital (0, 7, 257)-net over F256, using
- 1 times m-reduction [i] based on (1, 8, 257)-net in base 128, using
- (3, 16, 289)-net in base 128, using
- base change [i] based on digital (1, 14, 289)-net over F256, using
- net from sequence [i] based on digital (1, 288)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 1 and N(F) ≥ 289, using
- net from sequence [i] based on digital (1, 288)-sequence over F256, using
- base change [i] based on digital (1, 14, 289)-net over F256, using
- (1, 7, 257)-net in base 128, using
(23−13, 23, 1255607)-Net in Base 128 — Upper bound on s
There is no (10, 23, 1255608)-net in base 128, because
- 1 times m-reduction [i] would yield (10, 22, 1255608)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 22835 975071 427281 134915 632640 079897 869210 400075 > 12822 [i]