Best Known (21−12, 21, s)-Nets in Base 128
(21−12, 21, 321)-Net over F128 — Constructive and digital
Digital (9, 21, 321)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (3, 15, 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 (0, 6, 129)-net over F128, using
(21−12, 21, 427)-Net over F128 — Digital
Digital (9, 21, 427)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12821, 427, F128, 12) (dual of [427, 406, 13]-code), using
- 41 step Varšamov–Edel lengthening with (ri) = (1, 40 times 0) [i] based on linear OA(12820, 385, F128, 12) (dual of [385, 365, 13]-code), using
- construction XX applied to C1 = C([380,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([380,10]) [i] based on
- linear OA(12818, 381, F128, 11) (dual of [381, 363, 12]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12818, 381, F128, 11) (dual of [381, 363, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12820, 381, F128, 12) (dual of [381, 361, 13]-code), using the BCH-code C(I) with length 381 | 1282−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12816, 381, F128, 10) (dual of [381, 365, 11]-code), using the expurgated narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [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([380,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([380,10]) [i] based on
- 41 step Varšamov–Edel lengthening with (ri) = (1, 40 times 0) [i] based on linear OA(12820, 385, F128, 12) (dual of [385, 365, 13]-code), using
(21−12, 21, 514)-Net in Base 128 — Constructive
(9, 21, 514)-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
- (2, 14, 257)-net in base 128, using
- 2 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on digital (0, 14, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 14, 257)-net over F256, using
- 2 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- (1, 7, 257)-net in base 128, using
(21−12, 21, 559308)-Net in Base 128 — Upper bound on s
There is no (9, 21, 559309)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 178 407268 112543 126709 159968 232408 759454 894460 > 12821 [i]