Best Known (209−28, 209, s)-Nets in Base 4
(209−28, 209, 18742)-Net over F4 — Constructive and digital
Digital (181, 209, 18742)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (162, 190, 18725)-net over F4, using
- net defined by OOA [i] based on linear OOA(4190, 18725, F4, 28, 28) (dual of [(18725, 28), 524110, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4190, 262150, F4, 28) (dual of [262150, 261960, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4190, 262153, F4, 28) (dual of [262153, 261963, 29]-code), using
- 1 times truncation [i] based on linear OA(4191, 262154, F4, 29) (dual of [262154, 261963, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 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(28) ⊂ Ce(26) [i] based on
- 1 times truncation [i] based on linear OA(4191, 262154, F4, 29) (dual of [262154, 261963, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4190, 262153, F4, 28) (dual of [262153, 261963, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(4190, 262150, F4, 28) (dual of [262150, 261960, 29]-code), using
- net defined by OOA [i] based on linear OOA(4190, 18725, F4, 28, 28) (dual of [(18725, 28), 524110, 29]-NRT-code), using
- digital (5, 19, 17)-net over F4, using
(209−28, 209, 230469)-Net over F4 — Digital
Digital (181, 209, 230469)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4209, 230469, F4, 28) (dual of [230469, 230260, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 262217, F4, 28) (dual of [262217, 262008, 29]-code), using
- 6 times code embedding in larger space [i] based on linear OA(4203, 262211, F4, 28) (dual of [262211, 262008, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(413, 67, F4, 6) (dual of [67, 54, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(410, 64, F4, 5) (dual of [64, 54, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 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(5) ⊂ Ce(4) [i] based on
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- 6 times code embedding in larger space [i] based on linear OA(4203, 262211, F4, 28) (dual of [262211, 262008, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4209, 262217, F4, 28) (dual of [262217, 262008, 29]-code), using
(209−28, 209, large)-Net in Base 4 — Upper bound on s
There is no (181, 209, large)-net in base 4, because
- 26 times m-reduction [i] would yield (181, 183, large)-net in base 4, but