Best Known (31, 54, s)-Nets in Base 64
(31, 54, 513)-Net over F64 — Constructive and digital
Digital (31, 54, 513)-net over F64, using
- t-expansion [i] based on digital (28, 54, 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
(31, 54, 1489)-Net in Base 64 — Constructive
(31, 54, 1489)-net in base 64, using
- 641 times duplication [i] based on (30, 53, 1489)-net in base 64, using
- net defined by OOA [i] based on OOA(6453, 1489, S64, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(6453, 16380, S64, 23), using
- discarding factors based on OA(6453, 16386, S64, 23), using
- discarding parts of the base [i] based on linear OA(12845, 16386, F128, 23) (dual of [16386, 16341, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12843, 16384, F128, 22) (dual of [16384, 16341, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(12845, 16386, F128, 23) (dual of [16386, 16341, 24]-code), using
- discarding factors based on OA(6453, 16386, S64, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(6453, 16380, S64, 23), using
- net defined by OOA [i] based on OOA(6453, 1489, S64, 23, 23), using
(31, 54, 4450)-Net over F64 — Digital
Digital (31, 54, 4450)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6454, 4450, F64, 23) (dual of [4450, 4396, 24]-code), using
- 343 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 10 times 0, 1, 30 times 0, 1, 84 times 0, 1, 212 times 0) [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- 343 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 10 times 0, 1, 30 times 0, 1, 84 times 0, 1, 212 times 0) [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
(31, 54, large)-Net in Base 64 — Upper bound on s
There is no (31, 54, large)-net in base 64, because
- 21 times m-reduction [i] would yield (31, 33, large)-net in base 64, but