Best Known (50−21, 50, s)-Nets in Base 64
(50−21, 50, 513)-Net over F64 — Constructive and digital
Digital (29, 50, 513)-net over F64, using
- t-expansion [i] based on digital (28, 50, 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
(50−21, 50, 1639)-Net in Base 64 — Constructive
(29, 50, 1639)-net in base 64, using
- net defined by OOA [i] based on OOA(6450, 1639, S64, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6450, 16391, S64, 21), using
- 1 times code embedding in larger space [i] based on OA(6449, 16390, S64, 21), using
- discarding parts of the base [i] based on linear OA(12842, 16390, F128, 21) (dual of [16390, 16348, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding parts of the base [i] based on linear OA(12842, 16390, F128, 21) (dual of [16390, 16348, 22]-code), using
- 1 times code embedding in larger space [i] based on OA(6449, 16390, S64, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6450, 16391, S64, 21), using
(50−21, 50, 4668)-Net over F64 — Digital
Digital (29, 50, 4668)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6450, 4668, F64, 21) (dual of [4668, 4618, 22]-code), using
- 561 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 4 times 0, 1, 16 times 0, 1, 49 times 0, 1, 137 times 0, 1, 348 times 0) [i] based on linear OA(6441, 4098, F64, 21) (dual of [4098, 4057, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(20) ⊂ Ce(19) [i] based on
- 561 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 4 times 0, 1, 16 times 0, 1, 49 times 0, 1, 137 times 0, 1, 348 times 0) [i] based on linear OA(6441, 4098, F64, 21) (dual of [4098, 4057, 22]-code), using
(50−21, 50, large)-Net in Base 64 — Upper bound on s
There is no (29, 50, large)-net in base 64, because
- 19 times m-reduction [i] would yield (29, 31, large)-net in base 64, but