Best Known (26, 26+18, s)-Nets in Base 64
(26, 26+18, 520)-Net over F64 — Constructive and digital
Digital (26, 44, 520)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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 (17, 35, 455)-net over F64, using
- net defined by OOA [i] based on linear OOA(6435, 455, F64, 18, 18) (dual of [(455, 18), 8155, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6435, 4095, F64, 18) (dual of [4095, 4060, 19]-code), using
- net defined by OOA [i] based on linear OOA(6435, 455, F64, 18, 18) (dual of [(455, 18), 8155, 19]-NRT-code), using
- digital (0, 9, 65)-net over F64, using
(26, 26+18, 1821)-Net in Base 64 — Constructive
(26, 44, 1821)-net in base 64, using
- 1 times m-reduction [i] based on (26, 45, 1821)-net in base 64, using
- net defined by OOA [i] based on OOA(6445, 1821, S64, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(6445, 16390, S64, 19), using
- discarding parts of the base [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding parts of the base [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on OA(6445, 16390, S64, 19), using
- net defined by OOA [i] based on OOA(6445, 1821, S64, 19, 19), using
(26, 26+18, 5521)-Net over F64 — Digital
Digital (26, 44, 5521)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6444, 5521, F64, 18) (dual of [5521, 5477, 19]-code), using
- 1414 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 15 times 0, 1, 50 times 0, 1, 143 times 0, 1, 373 times 0, 1, 823 times 0) [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- 1414 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 15 times 0, 1, 50 times 0, 1, 143 times 0, 1, 373 times 0, 1, 823 times 0) [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
(26, 26+18, large)-Net in Base 64 — Upper bound on s
There is no (26, 44, large)-net in base 64, because
- 16 times m-reduction [i] would yield (26, 28, large)-net in base 64, but