Best Known (21, 53, s)-Nets in Base 128
(21, 53, 345)-Net over F128 — Constructive and digital
Digital (21, 53, 345)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 16, 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 (5, 37, 216)-net over F128, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 5 and N(F) ≥ 216, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- digital (0, 16, 129)-net over F128, using
(21, 53, 386)-Net in Base 128 — Constructive
(21, 53, 386)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 16, 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
- (5, 37, 257)-net in base 128, using
- 3 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- base change [i] based on digital (0, 35, 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, 35, 257)-net over F256, using
- 3 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- digital (0, 16, 129)-net over F128, using
(21, 53, 414)-Net over F128 — Digital
Digital (21, 53, 414)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12853, 414, F128, 32) (dual of [414, 361, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, 424, F128, 32) (dual of [424, 371, 33]-code), using
- 32 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 22 times 0) [i] based on linear OA(12847, 386, F128, 32) (dual of [386, 339, 33]-code), using
- extended algebraic-geometric code AGe(F,353P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- 32 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 22 times 0) [i] based on linear OA(12847, 386, F128, 32) (dual of [386, 339, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, 424, F128, 32) (dual of [424, 371, 33]-code), using
(21, 53, 513)-Net in Base 128
(21, 53, 513)-net in base 128, using
- t-expansion [i] based on (17, 53, 513)-net in base 128, using
- 19 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
- base change [i] based on digital (8, 63, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- base change [i] based on digital (8, 63, 513)-net over F256, using
- 19 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(21, 53, 511518)-Net in Base 128 — Upper bound on s
There is no (21, 53, 511519)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 4809 853555 810796 462557 161696 318242 597979 121503 173471 626313 759221 488165 388898 703888 366088 439422 681958 766318 648755 > 12853 [i]