Best Known (145−18, 145, s)-Nets in Base 4
(145−18, 145, 466035)-Net over F4 — Constructive and digital
Digital (127, 145, 466035)-net over F4, using
- 41 times duplication [i] based on digital (126, 144, 466035)-net over F4, using
- net defined by OOA [i] based on linear OOA(4144, 466035, F4, 18, 18) (dual of [(466035, 18), 8388486, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4144, 4194315, F4, 18) (dual of [4194315, 4194171, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- OA 9-folding and stacking [i] based on linear OA(4144, 4194315, F4, 18) (dual of [4194315, 4194171, 19]-code), using
- net defined by OOA [i] based on linear OOA(4144, 466035, F4, 18, 18) (dual of [(466035, 18), 8388486, 19]-NRT-code), using
(145−18, 145, 1398105)-Net over F4 — Digital
Digital (127, 145, 1398105)-net over F4, using
- 41 times duplication [i] based on digital (126, 144, 1398105)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4144, 1398105, F4, 3, 18) (dual of [(1398105, 3), 4194171, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4144, 4194315, F4, 18) (dual of [4194315, 4194171, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(4133, 4194304, F4, 17) (dual of [4194304, 4194171, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(4144, 4194315, F4, 18) (dual of [4194315, 4194171, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4144, 1398105, F4, 3, 18) (dual of [(1398105, 3), 4194171, 19]-NRT-code), using
(145−18, 145, large)-Net in Base 4 — Upper bound on s
There is no (127, 145, large)-net in base 4, because
- 16 times m-reduction [i] would yield (127, 129, large)-net in base 4, but