Best Known (78, 96, s)-Nets in Base 16
(78, 96, 116574)-Net over F16 — Constructive and digital
Digital (78, 96, 116574)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 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 (63, 81, 116509)-net over F16, using
- net defined by OOA [i] based on linear OOA(1681, 116509, F16, 18, 18) (dual of [(116509, 18), 2097081, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(1681, 1048581, F16, 18) (dual of [1048581, 1048500, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- OA 9-folding and stacking [i] based on linear OA(1681, 1048581, F16, 18) (dual of [1048581, 1048500, 19]-code), using
- net defined by OOA [i] based on linear OOA(1681, 116509, F16, 18, 18) (dual of [(116509, 18), 2097081, 19]-NRT-code), using
- digital (6, 15, 65)-net over F16, using
(78, 96, 233018)-Net in Base 16 — Constructive
(78, 96, 233018)-net in base 16, using
- base change [i] based on (46, 64, 233018)-net in base 64, using
- 641 times duplication [i] based on (45, 63, 233018)-net in base 64, using
- base change [i] based on digital (36, 54, 233018)-net over F128, using
- net defined by OOA [i] based on linear OOA(12854, 233018, F128, 18, 18) (dual of [(233018, 18), 4194270, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12854, 2097162, F128, 18) (dual of [2097162, 2097108, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12854, 2097163, F128, 18) (dual of [2097163, 2097109, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(12854, 2097163, F128, 18) (dual of [2097163, 2097109, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(12854, 2097162, F128, 18) (dual of [2097162, 2097108, 19]-code), using
- net defined by OOA [i] based on linear OOA(12854, 233018, F128, 18, 18) (dual of [(233018, 18), 4194270, 19]-NRT-code), using
- base change [i] based on digital (36, 54, 233018)-net over F128, using
- 641 times duplication [i] based on (45, 63, 233018)-net in base 64, using
(78, 96, 3017113)-Net over F16 — Digital
Digital (78, 96, 3017113)-net over F16, using
(78, 96, large)-Net in Base 16 — Upper bound on s
There is no (78, 96, large)-net in base 16, because
- 16 times m-reduction [i] would yield (78, 80, large)-net in base 16, but