Best Known (50, 50+38, s)-Nets in Base 64
(50, 50+38, 617)-Net over F64 — Constructive and digital
Digital (50, 88, 617)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 22, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (28, 66, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- digital (3, 22, 104)-net over F64, using
(50, 50+38, 862)-Net in Base 64 — Constructive
(50, 88, 862)-net in base 64, using
- net defined by OOA [i] based on OOA(6488, 862, S64, 38, 38), using
- OA 19-folding and stacking [i] based on OA(6488, 16378, S64, 38), using
- discarding factors based on OA(6488, 16386, S64, 38), using
- discarding parts of the base [i] based on linear OA(12875, 16386, F128, 38) (dual of [16386, 16311, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(12875, 16384, F128, 38) (dual of [16384, 16309, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(12873, 16384, F128, 37) (dual of [16384, 16311, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- discarding parts of the base [i] based on linear OA(12875, 16386, F128, 38) (dual of [16386, 16311, 39]-code), using
- discarding factors based on OA(6488, 16386, S64, 38), using
- OA 19-folding and stacking [i] based on OA(6488, 16378, S64, 38), using
(50, 50+38, 4736)-Net over F64 — Digital
Digital (50, 88, 4736)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6488, 4736, F64, 38) (dual of [4736, 4648, 39]-code), using
- 625 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 4 times 0, 1, 9 times 0, 1, 21 times 0, 1, 43 times 0, 1, 85 times 0, 1, 163 times 0, 1, 291 times 0) [i] based on linear OA(6475, 4098, F64, 38) (dual of [4098, 4023, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(6475, 4096, F64, 38) (dual of [4096, 4021, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(6473, 4096, F64, 37) (dual of [4096, 4023, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 625 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 4 times 0, 1, 9 times 0, 1, 21 times 0, 1, 43 times 0, 1, 85 times 0, 1, 163 times 0, 1, 291 times 0) [i] based on linear OA(6475, 4098, F64, 38) (dual of [4098, 4023, 39]-code), using
(50, 50+38, large)-Net in Base 64 — Upper bound on s
There is no (50, 88, large)-net in base 64, because
- 36 times m-reduction [i] would yield (50, 52, large)-net in base 64, but