Best Known (116−41, 116, s)-Nets in Base 16
(116−41, 116, 587)-Net over F16 — Constructive and digital
Digital (75, 116, 587)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 26, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (49, 90, 522)-net over F16, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F256, using
- digital (6, 26, 65)-net over F16, using
(116−41, 116, 612)-Net in Base 16 — Constructive
(75, 116, 612)-net in base 16, using
- (u, u+v)-construction [i] based on
- (14, 34, 98)-net in base 16, using
- 1 times m-reduction [i] based on (14, 35, 98)-net in base 16, using
- base change [i] based on digital (7, 28, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- base change [i] based on digital (7, 28, 98)-net over F32, using
- 1 times m-reduction [i] based on (14, 35, 98)-net in base 16, using
- digital (41, 82, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 41, 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, 41, 257)-net over F256, using
- (14, 34, 98)-net in base 16, using
(116−41, 116, 3626)-Net over F16 — Digital
Digital (75, 116, 3626)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16116, 3626, F16, 41) (dual of [3626, 3510, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(16116, 4104, F16, 41) (dual of [4104, 3988, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,19]) [i] based on
- linear OA(16115, 4097, F16, 41) (dual of [4097, 3982, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(16109, 4097, F16, 39) (dual of [4097, 3988, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [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 C([0,20]) ⊂ C([0,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(16116, 4104, F16, 41) (dual of [4104, 3988, 42]-code), using
(116−41, 116, 4644124)-Net in Base 16 — Upper bound on s
There is no (75, 116, 4644125)-net in base 16, because
- 1 times m-reduction [i] would yield (75, 115, 4644125)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 2 977137 116940 136118 260416 505329 922891 065866 528492 941380 332923 640914 369513 327118 245751 343590 791394 704253 215793 902361 302507 369462 397898 675001 > 16115 [i]