Best Known (79−19, 79, s)-Nets in Base 4
(79−19, 79, 1028)-Net over F4 — Constructive and digital
Digital (60, 79, 1028)-net over F4, using
- 1 times m-reduction [i] based on digital (60, 80, 1028)-net over F4, 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
(79−19, 79, 1151)-Net over F4 — Digital
Digital (60, 79, 1151)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(479, 1151, F4, 19) (dual of [1151, 1072, 20]-code), using
- 108 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 16 times 0, 1, 28 times 0, 1, 47 times 0) [i] based on linear OA(472, 1036, F4, 19) (dual of [1036, 964, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(471, 1025, F4, 19) (dual of [1025, 954, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(461, 1025, F4, 17) (dual of [1025, 964, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- 108 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 16 times 0, 1, 28 times 0, 1, 47 times 0) [i] based on linear OA(472, 1036, F4, 19) (dual of [1036, 964, 20]-code), using
(79−19, 79, 228281)-Net in Base 4 — Upper bound on s
There is no (60, 79, 228282)-net in base 4, because
- 1 times m-reduction [i] would yield (60, 78, 228282)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 91346 756155 436010 392715 781471 735914 877980 047773 > 478 [i]