Best Known (58, 58+31, s)-Nets in Base 16
(58, 58+31, 585)-Net over F16 — Constructive and digital
Digital (58, 89, 585)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 27, 71)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (3, 18, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (2, 9, 33)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (31, 62, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 31, 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
- trace code for nets [i] based on digital (0, 31, 257)-net over F256, using
- digital (12, 27, 71)-net over F16, using
(58, 58+31, 643)-Net in Base 16 — Constructive
(58, 89, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (12, 27, 129)-net in base 16, using
- 1 times m-reduction [i] based on (12, 28, 129)-net in base 16, using
- base change [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
- base change [i] based on digital (0, 16, 129)-net over F128, using
- 1 times m-reduction [i] based on (12, 28, 129)-net in base 16, using
- digital (31, 62, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 31, 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
- trace code for nets [i] based on digital (0, 31, 257)-net over F256, using
- (12, 27, 129)-net in base 16, using
(58, 58+31, 3492)-Net over F16 — Digital
Digital (58, 89, 3492)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1689, 3492, F16, 31) (dual of [3492, 3403, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(1689, 4103, F16, 31) (dual of [4103, 4014, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(1688, 4096, F16, 31) (dual of [4096, 4008, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(1682, 4096, F16, 29) (dual of [4096, 4014, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(1689, 4103, F16, 31) (dual of [4103, 4014, 32]-code), using
(58, 58+31, 4964154)-Net in Base 16 — Upper bound on s
There is no (58, 89, 4964155)-net in base 16, because
- 1 times m-reduction [i] would yield (58, 88, 4964155)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 9174 021925 691976 011992 215954 351713 813826 421755 171073 662418 970987 049407 495221 395769 623059 169707 517980 367376 > 1688 [i]