Best Known (255−48, 255, s)-Nets in Base 4
(255−48, 255, 1553)-Net over F4 — Constructive and digital
Digital (207, 255, 1553)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 27, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (180, 228, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 76, 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
- trace code for nets [i] based on digital (28, 76, 513)-net over F64, using
- digital (3, 27, 14)-net over F4, using
(255−48, 255, 12626)-Net over F4 — Digital
Digital (207, 255, 12626)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4255, 12626, F4, 48) (dual of [12626, 12371, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(4255, 16401, F4, 48) (dual of [16401, 16146, 49]-code), using
- construction XX applied to Ce(48) ⊂ Ce(45) ⊂ Ce(44) [i] based on
- linear OA(4253, 16384, F4, 49) (dual of [16384, 16131, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(4239, 16384, F4, 46) (dual of [16384, 16145, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 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(48) ⊂ Ce(45) ⊂ Ce(44) [i] based on
- discarding factors / shortening the dual code based on linear OA(4255, 16401, F4, 48) (dual of [16401, 16146, 49]-code), using
(255−48, 255, 8149441)-Net in Base 4 — Upper bound on s
There is no (207, 255, 8149442)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3351 956240 285778 274354 852055 500130 376367 143175 387907 924031 913065 469901 340524 288242 297918 860801 051146 631195 899318 995254 649083 624653 797193 016099 749899 127416 > 4255 [i]