Best Known (179, 205, s)-Nets in Base 4
(179, 205, 80669)-Net over F4 — Constructive and digital
Digital (179, 205, 80669)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (165, 191, 80660)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 80660, F4, 26, 26) (dual of [(80660, 26), 2096969, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4191, 1048580, F4, 26) (dual of [1048580, 1048389, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4191, 1048580, F4, 26) (dual of [1048580, 1048389, 27]-code), using
- net defined by OOA [i] based on linear OOA(4191, 80660, F4, 26, 26) (dual of [(80660, 26), 2096969, 27]-NRT-code), using
- digital (1, 14, 9)-net over F4, using
(179, 205, 524320)-Net over F4 — Digital
Digital (179, 205, 524320)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4205, 524320, F4, 2, 26) (dual of [(524320, 2), 1048435, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4205, 1048640, F4, 26) (dual of [1048640, 1048435, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4204, 1048639, F4, 26) (dual of [1048639, 1048435, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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]
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4204, 1048639, F4, 26) (dual of [1048639, 1048435, 27]-code), using
- OOA 2-folding [i] based on linear OA(4205, 1048640, F4, 26) (dual of [1048640, 1048435, 27]-code), using
(179, 205, large)-Net in Base 4 — Upper bound on s
There is no (179, 205, large)-net in base 4, because
- 24 times m-reduction [i] would yield (179, 181, large)-net in base 4, but