Best Known (34, 51, s)-Nets in Base 4
(34, 51, 195)-Net over F4 — Constructive and digital
Digital (34, 51, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 17, 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
(34, 51, 208)-Net over F4 — Digital
Digital (34, 51, 208)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(451, 208, F4, 17) (dual of [208, 157, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(451, 263, F4, 17) (dual of [263, 212, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- linear OA(449, 256, F4, 17) (dual of [256, 207, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(445, 256, F4, 15) (dual of [256, 211, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(441, 256, F4, 14) (dual of [256, 215, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(451, 263, F4, 17) (dual of [263, 212, 18]-code), using
(34, 51, 7262)-Net in Base 4 — Upper bound on s
There is no (34, 51, 7263)-net in base 4, because
- 1 times m-reduction [i] would yield (34, 50, 7263)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 268602 419286 252192 540073 152645 > 450 [i]