Best Known (20−11, 20, s)-Nets in Base 128
(20−11, 20, 408)-Net over F128 — Constructive and digital
Digital (9, 20, 408)-net over F128, using
- generalized (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 (0, 5, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (1, 12, 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
(20−11, 20, 515)-Net in Base 128 — Constructive
(9, 20, 515)-net in base 128, using
- (u, u+v)-construction [i] based on
- (1, 6, 257)-net in base 128, using
- 2 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
- 2 times m-reduction [i] based on (1, 8, 257)-net in base 128, using
- (3, 14, 258)-net in base 128, using
- 2 times m-reduction [i] based on (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
- 2 times m-reduction [i] based on (3, 16, 258)-net in base 128, using
- (1, 6, 257)-net in base 128, using
(20−11, 20, 590)-Net over F128 — Digital
Digital (9, 20, 590)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12820, 590, F128, 11) (dual of [590, 570, 12]-code), using
- 204 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0, 1, 186 times 0) [i] based on linear OA(12818, 384, F128, 11) (dual of [384, 366, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(12818, 382, F128, 11) (dual of [382, 364, 12]-code), using an extension Ce(10) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12816, 382, F128, 10) (dual of [382, 366, 11]-code), using an extension Ce(9) of the narrow-sense BCH-code C(I) with length 381 | 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 204 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0, 1, 186 times 0) [i] based on linear OA(12818, 384, F128, 11) (dual of [384, 366, 12]-code), using
(20−11, 20, 2086555)-Net in Base 128 — Upper bound on s
There is no (9, 20, 2086556)-net in base 128, because
- 1 times m-reduction [i] would yield (9, 19, 2086556)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 10889 042832 991671 701373 147708 865919 888070 > 12819 [i]