Best Known (31−10, 31, s)-Nets in Base 4
(31−10, 31, 195)-Net over F4 — Constructive and digital
Digital (21, 31, 195)-net over F4, using
- 41 times duplication [i] based on digital (20, 30, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 10, 65)-net over F64, using
(31−10, 31, 222)-Net over F4 — Digital
Digital (21, 31, 222)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(431, 222, F4, 10) (dual of [222, 191, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(431, 263, F4, 10) (dual of [263, 232, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(8) ⊂ Ce(6) [i] based on
- linear OA(429, 256, F4, 10) (dual of [256, 227, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(425, 256, F4, 9) (dual of [256, 231, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(421, 256, F4, 7) (dual of [256, 235, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(9) ⊂ Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(431, 263, F4, 10) (dual of [263, 232, 11]-code), using
(31−10, 31, 4689)-Net in Base 4 — Upper bound on s
There is no (21, 31, 4690)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 4 613023 955524 745860 > 431 [i]