Best Known (31, 53, s)-Nets in Base 64
(31, 53, 513)-Net over F64 — Constructive and digital
Digital (31, 53, 513)-net over F64, using
- t-expansion [i] based on digital (28, 53, 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, 53, 1490)-Net in Base 64 — Constructive
(31, 53, 1490)-net in base 64, using
- net defined by OOA [i] based on OOA(6453, 1490, S64, 22, 22), using
- OA 11-folding and stacking [i] based on OA(6453, 16390, S64, 22), using
- discarding factors based on OA(6453, 16392, S64, 22), using
- discarding parts of the base [i] based on linear OA(12845, 16392, F128, 22) (dual of [16392, 16347, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- 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(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding parts of the base [i] based on linear OA(12845, 16392, F128, 22) (dual of [16392, 16347, 23]-code), using
- discarding factors based on OA(6453, 16392, S64, 22), using
- OA 11-folding and stacking [i] based on OA(6453, 16390, S64, 22), using
(31, 53, 5110)-Net over F64 — Digital
Digital (31, 53, 5110)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6453, 5110, F64, 22) (dual of [5110, 5057, 23]-code), using
- 1000 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 11 times 0, 1, 35 times 0, 1, 99 times 0, 1, 261 times 0, 1, 585 times 0) [i] based on linear OA(6444, 4101, F64, 22) (dual of [4101, 4057, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- 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(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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 Ce(21) ⊂ Ce(19) [i] based on
- 1000 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 11 times 0, 1, 35 times 0, 1, 99 times 0, 1, 261 times 0, 1, 585 times 0) [i] based on linear OA(6444, 4101, F64, 22) (dual of [4101, 4057, 23]-code), using
(31, 53, large)-Net in Base 64 — Upper bound on s
There is no (31, 53, large)-net in base 64, because
- 20 times m-reduction [i] would yield (31, 33, large)-net in base 64, but