Best Known (112, 112+18, s)-Nets in Base 4
(112, 112+18, 29142)-Net over F4 — Constructive and digital
Digital (112, 130, 29142)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (100, 118, 29128)-net over F4, using
- net defined by OOA [i] based on linear OOA(4118, 29128, F4, 18, 18) (dual of [(29128, 18), 524186, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4118, 262152, F4, 18) (dual of [262152, 262034, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4118, 262153, F4, 18) (dual of [262153, 262035, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4118, 262153, F4, 18) (dual of [262153, 262035, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4118, 262152, F4, 18) (dual of [262152, 262034, 19]-code), using
- net defined by OOA [i] based on linear OOA(4118, 29128, F4, 18, 18) (dual of [(29128, 18), 524186, 19]-NRT-code), using
- digital (3, 12, 14)-net over F4, using
(112, 112+18, 161992)-Net over F4 — Digital
Digital (112, 130, 161992)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4130, 161992, F4, 18) (dual of [161992, 161862, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4130, 262192, F4, 18) (dual of [262192, 262062, 19]-code), using
- 5 times code embedding in larger space [i] based on linear OA(4125, 262187, F4, 18) (dual of [262187, 262062, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(4125, 262187, F4, 18) (dual of [262187, 262062, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4130, 262192, F4, 18) (dual of [262192, 262062, 19]-code), using
(112, 112+18, large)-Net in Base 4 — Upper bound on s
There is no (112, 130, large)-net in base 4, because
- 16 times m-reduction [i] would yield (112, 114, large)-net in base 4, but