Best Known (4, 10, s)-Nets in Base 128
(4, 10, 279)-Net over F128 — Constructive and digital
Digital (4, 10, 279)-net over F128, using
- (u, u+v)-construction [i] based on
- 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 (1, 7, 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 (0, 3, 129)-net over F128, using
(4, 10, 385)-Net over F128 — Digital
Digital (4, 10, 385)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12810, 385, F128, 6) (dual of [385, 375, 7]-code), using
- construction XX applied to C1 = C([380,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([380,4]) [i] based on
- linear OA(1288, 381, F128, 5) (dual of [381, 373, 6]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,1,2,3}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1288, 381, F128, 5) (dual of [381, 373, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12810, 381, F128, 6) (dual of [381, 371, 7]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1286, 381, F128, 4) (dual of [381, 375, 5]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 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
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([380,3]), C2 = C([0,4]), C3 = C1 + C2 = C([0,3]), and C∩ = C1 ∩ C2 = C([380,4]) [i] based on
(4, 10, 386)-Net in Base 128 — Constructive
(4, 10, 386)-net in base 128, using
- (u, u+v)-construction [i] based on
- 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
- (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
- digital (0, 3, 129)-net over F128, using
(4, 10, 151220)-Net in Base 128 — Upper bound on s
There is no (4, 10, 151221)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 1180 606170 292646 370092 > 12810 [i]