Best Known (59, 79, s)-Nets in Base 4
(59, 79, 514)-Net over F4 — Constructive and digital
Digital (59, 79, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(39,16) in PG(78,4)) for nets [i] based on digital (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 20, 257)-net over F256, using
(59, 79, 1010)-Net over F4 — Digital
Digital (59, 79, 1010)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(479, 1010, F4, 20) (dual of [1010, 931, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(479, 1042, F4, 20) (dual of [1042, 963, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(461, 1024, F4, 17) (dual of [1024, 963, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(43, 18, F4, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(479, 1042, F4, 20) (dual of [1042, 963, 21]-code), using
(59, 79, 86117)-Net in Base 4 — Upper bound on s
There is no (59, 79, 86118)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 365412 896540 477873 241289 379180 130628 890159 051880 > 479 [i]