Best Known (168, 168+23, s)-Nets in Base 4
(168, 168+23, 381302)-Net over F4 — Constructive and digital
Digital (168, 191, 381302)-net over F4, using
- 42 times duplication [i] based on digital (166, 189, 381302)-net over F4, using
- net defined by OOA [i] based on linear OOA(4189, 381302, F4, 23, 23) (dual of [(381302, 23), 8769757, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4189, 4194323, F4, 23) (dual of [4194323, 4194134, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 4194327, F4, 23) (dual of [4194327, 4194138, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(41, 23, F4, 1) (dual of [23, 22, 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 Ce(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4189, 4194327, F4, 23) (dual of [4194327, 4194138, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4189, 4194323, F4, 23) (dual of [4194323, 4194134, 24]-code), using
- net defined by OOA [i] based on linear OOA(4189, 381302, F4, 23, 23) (dual of [(381302, 23), 8769757, 24]-NRT-code), using
(168, 168+23, 1398110)-Net over F4 — Digital
Digital (168, 191, 1398110)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4191, 1398110, F4, 3, 23) (dual of [(1398110, 3), 4194139, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4191, 4194330, F4, 23) (dual of [4194330, 4194139, 24]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4189, 4194328, F4, 23) (dual of [4194328, 4194139, 24]-code), using
- construction X4 applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4166, 4194304, F4, 21) (dual of [4194304, 4194138, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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 X4 applied to Ce(22) ⊂ Ce(20) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4189, 4194328, F4, 23) (dual of [4194328, 4194139, 24]-code), using
- OOA 3-folding [i] based on linear OA(4191, 4194330, F4, 23) (dual of [4194330, 4194139, 24]-code), using
(168, 168+23, large)-Net in Base 4 — Upper bound on s
There is no (168, 191, large)-net in base 4, because
- 21 times m-reduction [i] would yield (168, 170, large)-net in base 4, but