Best Known (20, 20+15, s)-Nets in Base 64
(20, 20+15, 587)-Net over F64 — Constructive and digital
Digital (20, 35, 587)-net over F64, using
- 642 times duplication [i] based on digital (18, 33, 587)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 587, F64, 15, 15) (dual of [(587, 15), 8772, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6433, 4110, F64, 15) (dual of [4110, 4077, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [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]
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(6433, 4110, F64, 15) (dual of [4110, 4077, 16]-code), using
- net defined by OOA [i] based on linear OOA(6433, 587, F64, 15, 15) (dual of [(587, 15), 8772, 16]-NRT-code), using
(20, 20+15, 2341)-Net in Base 64 — Constructive
(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
(20, 20+15, 4262)-Net over F64 — Digital
Digital (20, 35, 4262)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6435, 4262, F64, 15) (dual of [4262, 4227, 16]-code), using
- 155 step Varšamov–Edel lengthening with (ri) = (3, 6 times 0, 1, 28 times 0, 1, 118 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
- 155 step Varšamov–Edel lengthening with (ri) = (3, 6 times 0, 1, 28 times 0, 1, 118 times 0) [i] based on linear OA(6430, 4102, F64, 15) (dual of [4102, 4072, 16]-code), using
(20, 20+15, 5463)-Net in Base 64
(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
(20, 20+15, large)-Net in Base 64 — Upper bound on s
There is no (20, 35, large)-net in base 64, because
- 13 times m-reduction [i] would yield (20, 22, large)-net in base 64, but