Best Known (51, 71, s)-Nets in Base 64
(51, 71, 26318)-Net over F64 — Constructive and digital
Digital (51, 71, 26318)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (38, 58, 26214)-net over F64, using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
- digital (3, 13, 104)-net over F64, using
(51, 71, 209716)-Net in Base 64 — Constructive
(51, 71, 209716)-net in base 64, using
- 641 times duplication [i] based on (50, 70, 209716)-net in base 64, 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
(51, 71, 706790)-Net over F64 — Digital
Digital (51, 71, 706790)-net over F64, using
(51, 71, 873861)-Net in Base 64
(51, 71, 873861)-net in base 64, using
- 641 times duplication [i] based on (50, 70, 873861)-net in base 64, using
- base change [i] based on digital (40, 60, 873861)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12860, 873861, F128, 2, 20) (dual of [(873861, 2), 1747662, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12860, 1048581, F128, 2, 20) (dual of [(1048581, 2), 2097102, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12860, 2097162, F128, 20) (dual of [2097162, 2097102, 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
- OOA 2-folding [i] based on linear OA(12860, 2097162, F128, 20) (dual of [2097162, 2097102, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(12860, 1048581, F128, 2, 20) (dual of [(1048581, 2), 2097102, 21]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12860, 873861, F128, 2, 20) (dual of [(873861, 2), 1747662, 21]-NRT-code), using
- base change [i] based on digital (40, 60, 873861)-net over F128, using
(51, 71, large)-Net in Base 64 — Upper bound on s
There is no (51, 71, large)-net in base 64, because
- 18 times m-reduction [i] would yield (51, 53, large)-net in base 64, but