Best Known (5, 5+7, s)-Nets in Base 128
(5, 5+7, 387)-Net over F128 — Constructive and digital
Digital (5, 12, 387)-net over F128, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 129)-net over F128, using
- digital (0, 3, 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, 7, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
(5, 5+7, 433)-Net over F128 — Digital
Digital (5, 12, 433)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12812, 433, F128, 7) (dual of [433, 421, 8]-code), using
- 49 step Varšamov–Edel lengthening with (ri) = (1, 48 times 0) [i] based on linear OA(12811, 383, F128, 7) (dual of [383, 372, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12811, 382, F128, 7) (dual of [382, 371, 8]-code), using an extension Ce(6) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12810, 382, F128, 6) (dual of [382, 372, 7]-code), using an extension Ce(5) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- 49 step Varšamov–Edel lengthening with (ri) = (1, 48 times 0) [i] based on linear OA(12811, 383, F128, 7) (dual of [383, 372, 8]-code), using
(5, 5+7, 514)-Net in Base 128 — Constructive
(5, 12, 514)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 16385)-net over F128, using
- net defined by OOA [i] based on linear OOA(1284, 16385, F128, 3, 3) (dual of [(16385, 3), 49151, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(1284, 16385, F128, 2, 3) (dual of [(16385, 2), 32766, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1284, 16385, F128, 3, 3) (dual of [(16385, 3), 49151, 4]-NRT-code), using
- (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
- digital (1, 4, 16385)-net over F128, using
(5, 5+7, 762107)-Net in Base 128 — Upper bound on s
There is no (5, 12, 762108)-net in base 128, because
- 1 times m-reduction [i] would yield (5, 11, 762108)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 151116 269533 909591 709091 > 12811 [i]