Best Known (222−40, 222, s)-Nets in Base 4
(222−40, 222, 1539)-Net over F4 — Constructive and digital
Digital (182, 222, 1539)-net over F4, using
- 9 times m-reduction [i] based on digital (182, 231, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 77, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 77, 513)-net over F64, using
(222−40, 222, 15860)-Net over F4 — Digital
Digital (182, 222, 15860)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4222, 15860, F4, 40) (dual of [15860, 15638, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, 16430, F4, 40) (dual of [16430, 16208, 41]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4221, 16429, F4, 40) (dual of [16429, 16208, 41]-code), using
- construction X applied to Ce(40) ⊂ Ce(33) [i] based on
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- 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(410, 45, F4, 5) (dual of [45, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(40) ⊂ Ce(33) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4221, 16429, F4, 40) (dual of [16429, 16208, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, 16430, F4, 40) (dual of [16430, 16208, 41]-code), using
(222−40, 222, large)-Net in Base 4 — Upper bound on s
There is no (182, 222, large)-net in base 4, because
- 38 times m-reduction [i] would yield (182, 184, large)-net in base 4, but