Best Known (156−22, 156, s)-Nets in Base 4
(156−22, 156, 23837)-Net over F4 — Constructive and digital
Digital (134, 156, 23837)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (123, 145, 23832)-net over F4, using
- net defined by OOA [i] based on linear OOA(4145, 23832, F4, 22, 22) (dual of [(23832, 22), 524159, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4145, 262152, F4, 22) (dual of [262152, 262007, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(40, 9, F4, 0) (dual of [9, 9, 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(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4145, 262152, F4, 22) (dual of [262152, 262007, 23]-code), using
- net defined by OOA [i] based on linear OOA(4145, 23832, F4, 22, 22) (dual of [(23832, 22), 524159, 23]-NRT-code), using
- digital (0, 11, 5)-net over F4, using
(156−22, 156, 131096)-Net over F4 — Digital
Digital (134, 156, 131096)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4156, 131096, F4, 2, 22) (dual of [(131096, 2), 262036, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4156, 262192, F4, 22) (dual of [262192, 262036, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4156, 262193, F4, 22) (dual of [262193, 262037, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4152, 262187, F4, 22) (dual of [262187, 262035, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(4152, 262189, F4, 19) (dual of [262189, 262037, 20]-code), using Gilbert–Varšamov bound and bm = 4152 > Vbs−1(k−1) = 2 073140 130308 252451 203559 837483 958671 321221 068454 777134 762413 288439 014062 930148 128428 833934 [i]
- linear OA(42, 4, F4, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,4)), using
- Reed–Solomon code RS(2,4) [i]
- linear OA(4152, 262187, F4, 22) (dual of [262187, 262035, 23]-code), using
- construction X with Varšamov bound [i] based on
- discarding factors / shortening the dual code based on linear OA(4156, 262193, F4, 22) (dual of [262193, 262037, 23]-code), using
- OOA 2-folding [i] based on linear OA(4156, 262192, F4, 22) (dual of [262192, 262036, 23]-code), using
(156−22, 156, large)-Net in Base 4 — Upper bound on s
There is no (134, 156, large)-net in base 4, because
- 20 times m-reduction [i] would yield (134, 136, large)-net in base 4, but