Best Known (28, 48, s)-Nets in Base 64
(28, 48, 513)-Net over F64 — Constructive and digital
Digital (28, 48, 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
(28, 48, 1639)-Net in Base 64 — Constructive
(28, 48, 1639)-net in base 64, using
- net defined by OOA [i] based on OOA(6448, 1639, S64, 20, 20), using
- OA 10-folding and stacking [i] based on OA(6448, 16390, S64, 20), using
- discarding factors based on OA(6448, 16392, S64, 20), using
- discarding parts of the base [i] based on linear OA(12841, 16392, F128, 20) (dual of [16392, 16351, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12833, 16384, F128, 17) (dual of [16384, 16351, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(16) [i] based on
- discarding parts of the base [i] based on linear OA(12841, 16392, F128, 20) (dual of [16392, 16351, 21]-code), using
- discarding factors based on OA(6448, 16392, S64, 20), using
- OA 10-folding and stacking [i] based on OA(6448, 16390, S64, 20), using
(28, 48, 4869)-Net over F64 — Digital
Digital (28, 48, 4869)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6448, 4869, F64, 20) (dual of [4869, 4821, 21]-code), using
- 762 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 70 times 0, 1, 189 times 0, 1, 465 times 0) [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 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(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- 762 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 70 times 0, 1, 189 times 0, 1, 465 times 0) [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
(28, 48, large)-Net in Base 64 — Upper bound on s
There is no (28, 48, large)-net in base 64, because
- 18 times m-reduction [i] would yield (28, 30, large)-net in base 64, but