Best Known (179, 179+35, s)-Nets in Base 4
(179, 179+35, 3856)-Net over F4 — Constructive and digital
Digital (179, 214, 3856)-net over F4, using
- 44 times duplication [i] based on digital (175, 210, 3856)-net over F4, using
- net defined by OOA [i] based on linear OOA(4210, 3856, F4, 35, 35) (dual of [(3856, 35), 134750, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4210, 65553, F4, 35) (dual of [65553, 65343, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4210, 65554, F4, 35) (dual of [65554, 65344, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- 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(41, 17, F4, 1) (dual of [17, 16, 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
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4210, 65554, F4, 35) (dual of [65554, 65344, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4210, 65553, F4, 35) (dual of [65553, 65343, 36]-code), using
- net defined by OOA [i] based on linear OOA(4210, 3856, F4, 35, 35) (dual of [(3856, 35), 134750, 36]-NRT-code), using
(179, 179+35, 33722)-Net over F4 — Digital
Digital (179, 214, 33722)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4214, 33722, F4, 35) (dual of [33722, 33508, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4214, 65565, F4, 35) (dual of [65565, 65351, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4214, 65565, F4, 35) (dual of [65565, 65351, 36]-code), using
(179, 179+35, large)-Net in Base 4 — Upper bound on s
There is no (179, 214, large)-net in base 4, because
- 33 times m-reduction [i] would yield (179, 181, large)-net in base 4, but