Best Known (179, 211, s)-Nets in Base 4
(179, 211, 4110)-Net over F4 — Constructive and digital
Digital (179, 211, 4110)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 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 (160, 192, 4096)-net over F4, using
- net defined by OOA [i] based on linear OOA(4192, 4096, F4, 32, 32) (dual of [(4096, 32), 130880, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(4192, 65536, F4, 32) (dual of [65536, 65344, 33]-code), using
- 1 times truncation [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using
- OA 16-folding and stacking [i] based on linear OA(4192, 65536, F4, 32) (dual of [65536, 65344, 33]-code), using
- net defined by OOA [i] based on linear OOA(4192, 4096, F4, 32, 32) (dual of [(4096, 32), 130880, 33]-NRT-code), using
- digital (3, 19, 14)-net over F4, using
(179, 211, 65607)-Net over F4 — Digital
Digital (179, 211, 65607)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4211, 65607, F4, 32) (dual of [65607, 65396, 33]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4206, 65598, F4, 32) (dual of [65598, 65392, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(413, 61, F4, 6) (dual of [61, 48, 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([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(4206, 65602, F4, 30) (dual of [65602, 65396, 31]-code), using Gilbert–Varšamov bound and bm = 4206 > Vbs−1(k−1) = 378 230648 133154 475780 207693 252119 461232 352922 298145 706430 238006 436037 430799 117165 068569 623147 008411 713569 229202 748836 607412 [i]
- linear OA(41, 5, F4, 1) (dual of [5, 4, 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(4206, 65598, F4, 32) (dual of [65598, 65392, 33]-code), using
- construction X with Varšamov bound [i] based on
(179, 211, large)-Net in Base 4 — Upper bound on s
There is no (179, 211, large)-net in base 4, because
- 30 times m-reduction [i] would yield (179, 181, large)-net in base 4, but