Best Known (41, 41+31, s)-Nets in Base 64
(41, 41+31, 513)-Net over F64 — Constructive and digital
Digital (41, 72, 513)-net over F64, using
- t-expansion [i] based on digital (28, 72, 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
(41, 41+31, 1092)-Net in Base 64 — Constructive
(41, 72, 1092)-net in base 64, using
- net defined by OOA [i] based on OOA(6472, 1092, S64, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(6472, 16381, S64, 31), using
- discarding factors based on OA(6472, 16386, S64, 31), using
- discarding parts of the base [i] based on linear OA(12861, 16386, F128, 31) (dual of [16386, 16325, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- discarding parts of the base [i] based on linear OA(12861, 16386, F128, 31) (dual of [16386, 16325, 32]-code), using
- discarding factors based on OA(6472, 16386, S64, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(6472, 16381, S64, 31), using
(41, 41+31, 4476)-Net over F64 — Digital
Digital (41, 72, 4476)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6472, 4476, F64, 31) (dual of [4476, 4404, 32]-code), using
- 364 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 6 times 0, 1, 17 times 0, 1, 40 times 0, 1, 93 times 0, 1, 200 times 0) [i] based on linear OA(6462, 4102, F64, 31) (dual of [4102, 4040, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- linear OA(6461, 4097, F64, 31) (dual of [4097, 4036, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(6457, 4097, F64, 29) (dual of [4097, 4040, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,15]) ⊂ C([0,14]) [i] based on
- 364 step Varšamov–Edel lengthening with (ri) = (5, 0, 0, 1, 6 times 0, 1, 17 times 0, 1, 40 times 0, 1, 93 times 0, 1, 200 times 0) [i] based on linear OA(6462, 4102, F64, 31) (dual of [4102, 4040, 32]-code), using
(41, 41+31, large)-Net in Base 64 — Upper bound on s
There is no (41, 72, large)-net in base 64, because
- 29 times m-reduction [i] would yield (41, 43, large)-net in base 64, but