Best Known (146, 165, s)-Nets in Base 4
(146, 165, 466043)-Net over F4 — Constructive and digital
Digital (146, 165, 466043)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (136, 155, 466034)-net over F4, using
- net defined by OOA [i] based on linear OOA(4155, 466034, F4, 19, 19) (dual of [(466034, 19), 8854491, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4155, 4194307, F4, 19) (dual of [4194307, 4194152, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4155, 4194315, F4, 19) (dual of [4194315, 4194160, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4155, 4194304, F4, 19) (dual of [4194304, 4194149, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4155, 4194315, F4, 19) (dual of [4194315, 4194160, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4155, 4194307, F4, 19) (dual of [4194307, 4194152, 20]-code), using
- net defined by OOA [i] based on linear OOA(4155, 466034, F4, 19, 19) (dual of [(466034, 19), 8854491, 20]-NRT-code), using
- digital (1, 10, 9)-net over F4, using
(146, 165, 2097184)-Net over F4 — Digital
Digital (146, 165, 2097184)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4165, 2097184, F4, 2, 19) (dual of [(2097184, 2), 4194203, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4165, 4194368, F4, 19) (dual of [4194368, 4194203, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(4155, 4194304, F4, 19) (dual of [4194304, 4194149, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4100, 4194304, F4, 13) (dual of [4194304, 4194204, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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]
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(4165, 4194368, F4, 19) (dual of [4194368, 4194203, 20]-code), using
(146, 165, large)-Net in Base 4 — Upper bound on s
There is no (146, 165, large)-net in base 4, because
- 17 times m-reduction [i] would yield (146, 148, large)-net in base 4, but