Best Known (31, 31+12, s)-Nets in Base 64
(31, 31+12, 43821)-Net over F64 — Constructive and digital
Digital (31, 43, 43821)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 3, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (22, 34, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6434, 43691, F64, 12, 12) (dual of [(43691, 12), 524258, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6434, 262146, F64, 12) (dual of [262146, 262112, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6434, 262147, F64, 12) (dual of [262147, 262113, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(6434, 262147, F64, 12) (dual of [262147, 262113, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6434, 262146, F64, 12) (dual of [262146, 262112, 13]-code), using
- net defined by OOA [i] based on linear OOA(6434, 43691, F64, 12, 12) (dual of [(43691, 12), 524258, 13]-NRT-code), using
- digital (3, 9, 130)-net over F64, using
(31, 31+12, 349527)-Net in Base 64 — Constructive
(31, 43, 349527)-net in base 64, using
- 641 times duplication [i] based on (30, 42, 349527)-net in base 64, using
- base change [i] based on digital (24, 36, 349527)-net over F128, using
- net defined by OOA [i] based on linear OOA(12836, 349527, F128, 12, 12) (dual of [(349527, 12), 4194288, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12836, 2097162, F128, 12) (dual of [2097162, 2097126, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12836, 2097163, F128, 12) (dual of [2097163, 2097127, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(12836, 2097163, F128, 12) (dual of [2097163, 2097127, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(12836, 2097162, F128, 12) (dual of [2097162, 2097126, 13]-code), using
- net defined by OOA [i] based on linear OOA(12836, 349527, F128, 12, 12) (dual of [(349527, 12), 4194288, 13]-NRT-code), using
- base change [i] based on digital (24, 36, 349527)-net over F128, using
(31, 31+12, 895773)-Net over F64 — Digital
Digital (31, 43, 895773)-net over F64, using
(31, 31+12, 1048581)-Net in Base 64
(31, 43, 1048581)-net in base 64, using
- 641 times duplication [i] based on (30, 42, 1048581)-net in base 64, using
- base change [i] based on digital (24, 36, 1048581)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12836, 1048581, F128, 2, 12) (dual of [(1048581, 2), 2097126, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12836, 2097162, F128, 12) (dual of [2097162, 2097126, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12836, 2097163, F128, 12) (dual of [2097163, 2097127, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(11) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(12836, 2097163, F128, 12) (dual of [2097163, 2097127, 13]-code), using
- OOA 2-folding [i] based on linear OA(12836, 2097162, F128, 12) (dual of [2097162, 2097126, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12836, 1048581, F128, 2, 12) (dual of [(1048581, 2), 2097126, 13]-NRT-code), using
- base change [i] based on digital (24, 36, 1048581)-net over F128, using
(31, 31+12, large)-Net in Base 64 — Upper bound on s
There is no (31, 43, large)-net in base 64, because
- 10 times m-reduction [i] would yield (31, 33, large)-net in base 64, but