Best Known (18, 30, s)-Nets in Base 64
(18, 30, 763)-Net over F64 — Constructive and digital
Digital (18, 30, 763)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (11, 23, 683)-net over F64, using
- net defined by OOA [i] based on linear OOA(6423, 683, F64, 12, 12) (dual of [(683, 12), 8173, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(640, 2, F64, 0) (dual of [2, 2, 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
- OA 6-folding and stacking [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- net defined by OOA [i] based on linear OOA(6423, 683, F64, 12, 12) (dual of [(683, 12), 8173, 13]-NRT-code), using
- digital (1, 7, 80)-net over F64, using
(18, 30, 2732)-Net in Base 64 — Constructive
(18, 30, 2732)-net in base 64, using
- net defined by OOA [i] based on OOA(6430, 2732, S64, 12, 12), using
- OA 6-folding and stacking [i] based on OA(6430, 16392, S64, 12), using
- discarding parts of the base [i] based on linear OA(12825, 16392, F128, 12) (dual of [16392, 16367, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-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 parts of the base [i] based on linear OA(12825, 16392, F128, 12) (dual of [16392, 16367, 13]-code), using
- OA 6-folding and stacking [i] based on OA(6430, 16392, S64, 12), using
(18, 30, 6665)-Net over F64 — Digital
Digital (18, 30, 6665)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6430, 6665, F64, 12) (dual of [6665, 6635, 13]-code), using
- 2560 step Varšamov–Edel lengthening with (ri) = (3, 7 times 0, 1, 44 times 0, 1, 187 times 0, 1, 650 times 0, 1, 1667 times 0) [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(640, 2, F64, 0) (dual of [2, 2, 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
- 2560 step Varšamov–Edel lengthening with (ri) = (3, 7 times 0, 1, 44 times 0, 1, 187 times 0, 1, 650 times 0, 1, 1667 times 0) [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
(18, 30, large)-Net in Base 64 — Upper bound on s
There is no (18, 30, large)-net in base 64, because
- 10 times m-reduction [i] would yield (18, 20, large)-net in base 64, but