Best Known (107−20, 107, s)-Nets in Base 16
(107−20, 107, 104923)-Net over F16 — Constructive and digital
Digital (87, 107, 104923)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 16, 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 (71, 91, 104858)-net over F16, using
- net defined by OOA [i] based on linear OOA(1691, 104858, F16, 20, 20) (dual of [(104858, 20), 2097069, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1691, 1048580, F16, 20) (dual of [1048580, 1048489, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1691, 1048581, F16, 20) (dual of [1048581, 1048490, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(1691, 1048581, F16, 20) (dual of [1048581, 1048490, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(1691, 1048580, F16, 20) (dual of [1048580, 1048489, 21]-code), using
- net defined by OOA [i] based on linear OOA(1691, 104858, F16, 20, 20) (dual of [(104858, 20), 2097069, 21]-NRT-code), using
- digital (6, 16, 65)-net over F16, using
(107−20, 107, 209716)-Net in Base 16 — Constructive
(87, 107, 209716)-net in base 16, using
- 162 times duplication [i] based on (85, 105, 209716)-net in base 16, using
- base change [i] based on digital (40, 60, 209716)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 209716, F128, 20, 20) (dual of [(209716, 20), 4194260, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12860, 2097160, F128, 20) (dual of [2097160, 2097100, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 2097163, F128, 20) (dual of [2097163, 2097103, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 2097163, F128, 20) (dual of [2097163, 2097103, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12860, 2097160, F128, 20) (dual of [2097160, 2097100, 21]-code), using
- net defined by OOA [i] based on linear OOA(12860, 209716, F128, 20, 20) (dual of [(209716, 20), 4194260, 21]-NRT-code), using
- base change [i] based on digital (40, 60, 209716)-net over F128, using
(107−20, 107, 3193172)-Net over F16 — Digital
Digital (87, 107, 3193172)-net over F16, using
(107−20, 107, large)-Net in Base 16 — Upper bound on s
There is no (87, 107, large)-net in base 16, because
- 18 times m-reduction [i] would yield (87, 89, large)-net in base 16, but