Best Known (111, 111+18, s)-Nets in Base 4
(111, 111+18, 29138)-Net over F4 — Constructive and digital
Digital (111, 129, 29138)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-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 (2, 11, 10)-net over F4, using
(111, 111+18, 148546)-Net over F4 — Digital
Digital (111, 129, 148546)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4129, 148546, F4, 18) (dual of [148546, 148417, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4129, 262159, F4, 18) (dual of [262159, 262030, 19]-code), using
- (u, u+v)-construction [i] based on
- linear OA(411, 15, F4, 9) (dual of [15, 4, 10]-code), using
- 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]
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4129, 262159, F4, 18) (dual of [262159, 262030, 19]-code), using
(111, 111+18, large)-Net in Base 4 — Upper bound on s
There is no (111, 129, large)-net in base 4, because
- 16 times m-reduction [i] would yield (111, 113, large)-net in base 4, but