Best Known (35, 35+26, s)-Nets in Base 64
(35, 35+26, 513)-Net over F64 — Constructive and digital
Digital (35, 61, 513)-net over F64, using
- t-expansion [i] based on digital (28, 61, 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
(35, 35+26, 1260)-Net in Base 64 — Constructive
(35, 61, 1260)-net in base 64, using
- 1 times m-reduction [i] based on (35, 62, 1260)-net in base 64, using
- net defined by OOA [i] based on OOA(6462, 1260, S64, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6462, 16381, S64, 27), using
- discarding factors based on OA(6462, 16386, S64, 27), using
- discarding parts of the base [i] based on linear OA(12853, 16386, F128, 27) (dual of [16386, 16333, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] 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]
- linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- discarding parts of the base [i] based on linear OA(12853, 16386, F128, 27) (dual of [16386, 16333, 28]-code), using
- discarding factors based on OA(6462, 16386, S64, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(6462, 16381, S64, 27), using
- net defined by OOA [i] based on OOA(6462, 1260, S64, 27, 27), using
(35, 35+26, 4546)-Net over F64 — Digital
Digital (35, 61, 4546)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6461, 4546, F64, 26) (dual of [4546, 4485, 27]-code), using
- 438 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 7 times 0, 1, 21 times 0, 1, 48 times 0, 1, 112 times 0, 1, 242 times 0) [i] based on linear OA(6451, 4098, F64, 26) (dual of [4098, 4047, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 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(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- 438 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 7 times 0, 1, 21 times 0, 1, 48 times 0, 1, 112 times 0, 1, 242 times 0) [i] based on linear OA(6451, 4098, F64, 26) (dual of [4098, 4047, 27]-code), using
(35, 35+26, large)-Net in Base 64 — Upper bound on s
There is no (35, 61, large)-net in base 64, because
- 24 times m-reduction [i] would yield (35, 37, large)-net in base 64, but