Best Known (64−27, 64, s)-Nets in Base 64
(64−27, 64, 513)-Net over F64 — Constructive and digital
Digital (37, 64, 513)-net over F64, using
- t-expansion [i] based on digital (28, 64, 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
(64−27, 64, 1260)-Net in Base 64 — Constructive
(37, 64, 1260)-net in base 64, using
- 641 times duplication [i] based on (36, 63, 1260)-net in base 64, using
- base change [i] based on digital (27, 54, 1260)-net over F128, using
- 1281 times duplication [i] based on digital (26, 53, 1260)-net over F128, using
- net defined by OOA [i] based on linear OOA(12853, 1260, F128, 27, 27) (dual of [(1260, 27), 33967, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12853, 16381, F128, 27) (dual of [16381, 16328, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12853, 16381, F128, 27) (dual of [16381, 16328, 28]-code), using
- net defined by OOA [i] based on linear OOA(12853, 1260, F128, 27, 27) (dual of [(1260, 27), 33967, 28]-NRT-code), using
- 1281 times duplication [i] based on digital (26, 53, 1260)-net over F128, using
- base change [i] based on digital (27, 54, 1260)-net over F128, using
(64−27, 64, 4862)-Net over F64 — Digital
Digital (37, 64, 4862)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6464, 4862, F64, 27) (dual of [4862, 4798, 28]-code), using
- 753 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 38 times 0, 1, 90 times 0, 1, 198 times 0, 1, 396 times 0) [i] based on linear OA(6453, 4098, F64, 27) (dual of [4098, 4045, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 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(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- 753 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 38 times 0, 1, 90 times 0, 1, 198 times 0, 1, 396 times 0) [i] based on linear OA(6453, 4098, F64, 27) (dual of [4098, 4045, 28]-code), using
(64−27, 64, large)-Net in Base 64 — Upper bound on s
There is no (37, 64, large)-net in base 64, because
- 25 times m-reduction [i] would yield (37, 39, large)-net in base 64, but