Best Known (55−33, 55, s)-Nets in Base 128
(55−33, 55, 384)-Net over F128 — Constructive and digital
Digital (22, 55, 384)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 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 (3, 36, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128 (see above)
- digital (3, 19, 192)-net over F128, using
(55−33, 55, 407)-Net in Base 128 — Constructive
(22, 55, 407)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 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
- (5, 38, 257)-net in base 128, using
- 2 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
- 2 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- digital (1, 17, 150)-net over F128, using
(55−33, 55, 445)-Net over F128 — Digital
Digital (22, 55, 445)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12855, 445, F128, 33) (dual of [445, 390, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 446, F128, 33) (dual of [446, 391, 34]-code), using
- 53 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 0, 1, 13 times 0, 1, 34 times 0) [i] based on linear OA(12848, 386, F128, 33) (dual of [386, 338, 34]-code), using
- extended algebraic-geometric code AGe(F,352P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- 53 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 0, 1, 13 times 0, 1, 34 times 0) [i] based on linear OA(12848, 386, F128, 33) (dual of [386, 338, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 446, F128, 33) (dual of [446, 391, 34]-code), using
(55−33, 55, 513)-Net in Base 128
(22, 55, 513)-net in base 128, using
- t-expansion [i] based on (17, 55, 513)-net in base 128, using
- 17 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
- 17 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(55−33, 55, 692729)-Net in Base 128 — Upper bound on s
There is no (22, 55, 692730)-net in base 128, because
- 1 times m-reduction [i] would yield (22, 54, 692730)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 615659 198299 670867 336860 628887 973766 346674 904864 093888 778437 994166 906438 565934 016645 132653 854675 542352 989931 230357 > 12854 [i]