Best Known (21, 21+14, s)-Nets in Base 64
(21, 21+14, 665)-Net over F64 — Constructive and digital
Digital (21, 35, 665)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (13, 27, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6427, 585, F64, 14, 14) (dual of [(585, 14), 8163, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
- net defined by OOA [i] based on linear OOA(6427, 585, F64, 14, 14) (dual of [(585, 14), 8163, 15]-NRT-code), using
- digital (1, 8, 80)-net over F64, using
(21, 21+14, 2342)-Net in Base 64 — Constructive
(21, 35, 2342)-net in base 64, using
- base change [i] based on digital (16, 30, 2342)-net over F128, using
- net defined by OOA [i] based on linear OOA(12830, 2342, F128, 14, 14) (dual of [(2342, 14), 32758, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12830, 16394, F128, 14) (dual of [16394, 16364, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16395, F128, 14) (dual of [16395, 16365, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16395, F128, 14) (dual of [16395, 16365, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12830, 16394, F128, 14) (dual of [16394, 16364, 15]-code), using
- net defined by OOA [i] based on linear OOA(12830, 2342, F128, 14, 14) (dual of [(2342, 14), 32758, 15]-NRT-code), using
(21, 21+14, 6576)-Net over F64 — Digital
Digital (21, 35, 6576)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6435, 6576, F64, 14) (dual of [6576, 6541, 15]-code), using
- 2466 step Varšamov–Edel lengthening with (ri) = (2, 1, 15 times 0, 1, 123 times 0, 1, 672 times 0, 1, 1651 times 0) [i] based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- 2466 step Varšamov–Edel lengthening with (ri) = (2, 1, 15 times 0, 1, 123 times 0, 1, 672 times 0, 1, 1651 times 0) [i] based on linear OA(6429, 4104, F64, 14) (dual of [4104, 4075, 15]-code), using
(21, 21+14, 8197)-Net in Base 64
(21, 35, 8197)-net in base 64, using
- base change [i] based on digital (16, 30, 8197)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12830, 8197, F128, 2, 14) (dual of [(8197, 2), 16364, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12830, 16394, F128, 14) (dual of [16394, 16364, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16395, F128, 14) (dual of [16395, 16365, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16395, F128, 14) (dual of [16395, 16365, 15]-code), using
- OOA 2-folding [i] based on linear OA(12830, 16394, F128, 14) (dual of [16394, 16364, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12830, 8197, F128, 2, 14) (dual of [(8197, 2), 16364, 15]-NRT-code), using
(21, 21+14, large)-Net in Base 64 — Upper bound on s
There is no (21, 35, large)-net in base 64, because
- 12 times m-reduction [i] would yield (21, 23, large)-net in base 64, but