Best Known (144, 178, s)-Nets in Base 4
(144, 178, 1118)-Net over F4 — Constructive and digital
Digital (144, 178, 1118)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (25, 42, 90)-net over F4, using
- trace code for nets [i] based on digital (4, 21, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- trace code for nets [i] based on digital (4, 21, 45)-net over F16, using
- digital (102, 136, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 34, 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, 34, 257)-net over F256, using
- digital (25, 42, 90)-net over F4, using
(144, 178, 9093)-Net over F4 — Digital
Digital (144, 178, 9093)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4178, 9093, F4, 34) (dual of [9093, 8915, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4178, 16394, F4, 34) (dual of [16394, 16216, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- linear OA(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(33) ⊂ Ce(32) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4178, 16394, F4, 34) (dual of [16394, 16216, 35]-code), using
(144, 178, 4816648)-Net in Base 4 — Upper bound on s
There is no (144, 178, 4816649)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 146784 012073 528760 410504 269306 132014 759080 988149 474745 989608 124642 578693 964939 439541 660461 284496 367468 289600 > 4178 [i]