Best Known (21, 21+15, s)-Nets in Base 64
(21, 21+15, 650)-Net over F64 — Constructive and digital
Digital (21, 36, 650)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 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 (14, 29, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- digital (0, 7, 65)-net over F64, using
(21, 21+15, 2341)-Net in Base 64 — Constructive
(21, 36, 2341)-net in base 64, using
- 641 times duplication [i] based on (20, 35, 2341)-net in base 64, using
- base change [i] based on digital (15, 30, 2341)-net over F128, using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
- base change [i] based on digital (15, 30, 2341)-net over F128, using
(21, 21+15, 4682)-Net over F64 — Digital
Digital (21, 36, 4682)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6436, 4682, F64, 15) (dual of [4682, 4646, 16]-code), using
- 574 step Varšamov–Edel lengthening with (ri) = (3, 6 times 0, 1, 28 times 0, 1, 118 times 0, 1, 418 times 0) [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 574 step Varšamov–Edel lengthening with (ri) = (3, 6 times 0, 1, 28 times 0, 1, 118 times 0, 1, 418 times 0) [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
(21, 21+15, 5463)-Net in Base 64
(21, 36, 5463)-net in base 64, using
- 641 times duplication [i] based on (20, 35, 5463)-net in base 64, using
- base change [i] based on digital (15, 30, 5463)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12830, 5463, F128, 3, 15) (dual of [(5463, 3), 16359, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12830, 16389, F128, 15) (dual of [16389, 16359, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- OOA 3-folding [i] based on linear OA(12830, 16389, F128, 15) (dual of [16389, 16359, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12830, 5463, F128, 3, 15) (dual of [(5463, 3), 16359, 16]-NRT-code), using
- base change [i] based on digital (15, 30, 5463)-net over F128, using
(21, 21+15, large)-Net in Base 64 — Upper bound on s
There is no (21, 36, large)-net in base 64, because
- 13 times m-reduction [i] would yield (21, 23, large)-net in base 64, but