Best Known (43, 43+33, s)-Nets in Base 64
(43, 43+33, 513)-Net over F64 — Constructive and digital
Digital (43, 76, 513)-net over F64, using
- t-expansion [i] based on digital (28, 76, 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
(43, 43+33, 1024)-Net in Base 64 — Constructive
(43, 76, 1024)-net in base 64, using
- net defined by OOA [i] based on OOA(6476, 1024, S64, 33, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(6476, 16385, S64, 33), using
- discarding factors based on OA(6476, 16386, S64, 33), using
- discarding parts of the base [i] based on linear OA(12865, 16386, F128, 33) (dual of [16386, 16321, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- linear OA(12865, 16384, F128, 33) (dual of [16384, 16319, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(32) ⊂ Ce(31) [i] based on
- discarding parts of the base [i] based on linear OA(12865, 16386, F128, 33) (dual of [16386, 16321, 34]-code), using
- discarding factors based on OA(6476, 16386, S64, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(6476, 16385, S64, 33), using
(43, 43+33, 4412)-Net over F64 — Digital
Digital (43, 76, 4412)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6476, 4412, F64, 33) (dual of [4412, 4336, 34]-code), using
- 303 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 16 times 0, 1, 36 times 0, 1, 79 times 0, 1, 159 times 0) [i] based on linear OA(6465, 4098, F64, 33) (dual of [4098, 4033, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- linear OA(6465, 4096, F64, 33) (dual of [4096, 4031, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(6463, 4096, F64, 32) (dual of [4096, 4033, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(32) ⊂ Ce(31) [i] based on
- 303 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 16 times 0, 1, 36 times 0, 1, 79 times 0, 1, 159 times 0) [i] based on linear OA(6465, 4098, F64, 33) (dual of [4098, 4033, 34]-code), using
(43, 43+33, large)-Net in Base 64 — Upper bound on s
There is no (43, 76, large)-net in base 64, because
- 31 times m-reduction [i] would yield (43, 45, large)-net in base 64, but