Best Known (183−25, 183, s)-Nets in Base 4
(183−25, 183, 87382)-Net over F4 — Constructive and digital
Digital (158, 183, 87382)-net over F4, using
- 41 times duplication [i] based on digital (157, 182, 87382)-net over F4, using
- net defined by OOA [i] based on linear OOA(4182, 87382, F4, 25, 25) (dual of [(87382, 25), 2184368, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4182, 1048585, F4, 25) (dual of [1048585, 1048403, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4182, 1048585, F4, 25) (dual of [1048585, 1048403, 26]-code), using
- net defined by OOA [i] based on linear OOA(4182, 87382, F4, 25, 25) (dual of [(87382, 25), 2184368, 26]-NRT-code), using
(183−25, 183, 349529)-Net over F4 — Digital
Digital (158, 183, 349529)-net over F4, using
- 41 times duplication [i] based on digital (157, 182, 349529)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4182, 349529, F4, 3, 25) (dual of [(349529, 3), 1048405, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- OOA 3-folding [i] based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4182, 349529, F4, 3, 25) (dual of [(349529, 3), 1048405, 26]-NRT-code), using
(183−25, 183, large)-Net in Base 4 — Upper bound on s
There is no (158, 183, large)-net in base 4, because
- 23 times m-reduction [i] would yield (158, 160, large)-net in base 4, but