Best Known (50, 50+37, s)-Nets in Base 64
(50, 50+37, 617)-Net over F64 — Constructive and digital
Digital (50, 87, 617)-net over F64, using
- 1 times m-reduction [i] based on 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
- (u, u+v)-construction [i] based on
(50, 50+37, 910)-Net in Base 64 — Constructive
(50, 87, 910)-net in base 64, using
- 641 times duplication [i] based on (49, 86, 910)-net in base 64, using
- net defined by OOA [i] based on OOA(6486, 910, S64, 37, 37), using
- OOA 18-folding and stacking with additional row [i] based on OA(6486, 16381, S64, 37), using
- discarding factors based on OA(6486, 16386, S64, 37), using
- discarding parts of the base [i] based on linear OA(12873, 16386, F128, 37) (dual of [16386, 16313, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(35) [i] based on
- 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(12871, 16384, F128, 36) (dual of [16384, 16313, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [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(36) ⊂ Ce(35) [i] based on
- discarding parts of the base [i] based on linear OA(12873, 16386, F128, 37) (dual of [16386, 16313, 38]-code), using
- discarding factors based on OA(6486, 16386, S64, 37), using
- OOA 18-folding and stacking with additional row [i] based on OA(6486, 16381, S64, 37), using
- net defined by OOA [i] based on OOA(6486, 910, S64, 37, 37), using
(50, 50+37, 5293)-Net over F64 — Digital
Digital (50, 87, 5293)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6487, 5293, F64, 37) (dual of [5293, 5206, 38]-code), using
- 1181 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 5 times 0, 1, 10 times 0, 1, 24 times 0, 1, 48 times 0, 1, 96 times 0, 1, 182 times 0, 1, 320 times 0, 1, 486 times 0) [i] based on linear OA(6473, 4098, F64, 37) (dual of [4098, 4025, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(35) [i] based on
- 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(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [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(36) ⊂ Ce(35) [i] based on
- 1181 step Varšamov–Edel lengthening with (ri) = (6, 0, 1, 5 times 0, 1, 10 times 0, 1, 24 times 0, 1, 48 times 0, 1, 96 times 0, 1, 182 times 0, 1, 320 times 0, 1, 486 times 0) [i] based on linear OA(6473, 4098, F64, 37) (dual of [4098, 4025, 38]-code), using
(50, 50+37, large)-Net in Base 64 — Upper bound on s
There is no (50, 87, large)-net in base 64, because
- 35 times m-reduction [i] would yield (50, 52, large)-net in base 64, but