Best Known (199, 199+26, s)-Nets in Base 4
(199, 199+26, 322649)-Net over F4 — Constructive and digital
Digital (199, 225, 322649)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 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 (184, 210, 322639)-net over F4, using
- net defined by OOA [i] based on linear OOA(4210, 322639, F4, 26, 26) (dual of [(322639, 26), 8388404, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4210, 4194307, F4, 26) (dual of [4194307, 4194097, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4210, 4194315, F4, 26) (dual of [4194315, 4194105, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4210, 4194304, F4, 26) (dual of [4194304, 4194094, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 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(4210, 4194315, F4, 26) (dual of [4194315, 4194105, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4210, 4194307, F4, 26) (dual of [4194307, 4194097, 27]-code), using
- net defined by OOA [i] based on linear OOA(4210, 322639, F4, 26, 26) (dual of [(322639, 26), 8388404, 27]-NRT-code), using
- digital (2, 15, 10)-net over F4, using
(199, 199+26, 2097192)-Net over F4 — Digital
Digital (199, 225, 2097192)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4225, 2097192, F4, 2, 26) (dual of [(2097192, 2), 4194159, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4225, 4194384, F4, 26) (dual of [4194384, 4194159, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4225, 4194385, F4, 26) (dual of [4194385, 4194160, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- linear OA(4210, 4194304, F4, 26) (dual of [4194304, 4194094, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(415, 81, F4, 7) (dual of [81, 66, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(415, 85, F4, 7) (dual of [85, 70, 8]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4225, 4194385, F4, 26) (dual of [4194385, 4194160, 27]-code), using
- OOA 2-folding [i] based on linear OA(4225, 4194384, F4, 26) (dual of [4194384, 4194159, 27]-code), using
(199, 199+26, large)-Net in Base 4 — Upper bound on s
There is no (199, 225, large)-net in base 4, because
- 24 times m-reduction [i] would yield (199, 201, large)-net in base 4, but