Best Known (118, 142, s)-Nets in Base 4
(118, 142, 1542)-Net over F4 — Constructive and digital
Digital (118, 142, 1542)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (34, 46, 514)-net over F4, using
- trace code for nets [i] based on digital (11, 23, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(11,256) in PG(22,16)) for nets [i] based on digital (0, 12, 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
- base reduction for projective spaces (embedding PG(11,256) in PG(22,16)) for nets [i] based on digital (0, 12, 257)-net over F256, using
- trace code for nets [i] based on digital (11, 23, 257)-net over F16, using
- digital (72, 96, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 24, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 24, 257)-net over F256, using
- digital (34, 46, 514)-net over F4, using
(118, 142, 16444)-Net over F4 — Digital
Digital (118, 142, 16444)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4142, 16444, F4, 24) (dual of [16444, 16302, 25]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4140, 16440, F4, 24) (dual of [16440, 16300, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4127, 16385, F4, 25) (dual of [16385, 16258, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(413, 55, F4, 6) (dual of [55, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4140, 16442, F4, 23) (dual of [16442, 16302, 24]-code), using Gilbert–Varšamov bound and bm = 4140 > Vbs−1(k−1) = 15502 846718 134182 428697 268842 629556 220490 060034 834207 484067 590474 526770 611148 573456 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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(4140, 16440, F4, 24) (dual of [16440, 16300, 25]-code), using
- construction X with Varšamov bound [i] based on
(118, 142, large)-Net in Base 4 — Upper bound on s
There is no (118, 142, large)-net in base 4, because
- 22 times m-reduction [i] would yield (118, 120, large)-net in base 4, but