Best Known (67−28, 67, s)-Nets in Base 64
(67−28, 67, 513)-Net over F64 — Constructive and digital
Digital (39, 67, 513)-net over F64, using
- t-expansion [i] based on digital (28, 67, 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
(67−28, 67, 1170)-Net in Base 64 — Constructive
(39, 67, 1170)-net in base 64, using
- t-expansion [i] based on (38, 67, 1170)-net in base 64, using
- net defined by OOA [i] based on OOA(6467, 1170, S64, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(6467, 16381, S64, 29), using
- discarding factors based on OA(6467, 16386, S64, 29), using
- discarding parts of the base [i] based on linear OA(12857, 16386, F128, 29) (dual of [16386, 16329, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- discarding parts of the base [i] based on linear OA(12857, 16386, F128, 29) (dual of [16386, 16329, 30]-code), using
- discarding factors based on OA(6467, 16386, S64, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(6467, 16381, S64, 29), using
- net defined by OOA [i] based on OOA(6467, 1170, S64, 29, 29), using
(67−28, 67, 5323)-Net over F64 — Digital
Digital (39, 67, 5323)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6467, 5323, F64, 28) (dual of [5323, 5256, 29]-code), using
- 1213 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 5 times 0, 1, 12 times 0, 1, 31 times 0, 1, 73 times 0, 1, 162 times 0, 1, 334 times 0, 1, 587 times 0) [i] based on linear OA(6455, 4098, F64, 28) (dual of [4098, 4043, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(6455, 4096, F64, 28) (dual of [4096, 4041, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(6453, 4096, F64, 27) (dual of [4096, 4043, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(27) ⊂ Ce(26) [i] based on
- 1213 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 5 times 0, 1, 12 times 0, 1, 31 times 0, 1, 73 times 0, 1, 162 times 0, 1, 334 times 0, 1, 587 times 0) [i] based on linear OA(6455, 4098, F64, 28) (dual of [4098, 4043, 29]-code), using
(67−28, 67, large)-Net in Base 64 — Upper bound on s
There is no (39, 67, large)-net in base 64, because
- 26 times m-reduction [i] would yield (39, 41, large)-net in base 64, but