Best Known (259−34, 259, s)-Nets in Base 4
(259−34, 259, 61683)-Net over F4 — Constructive and digital
Digital (225, 259, 61683)-net over F4, using
- 43 times duplication [i] based on digital (222, 256, 61683)-net over F4, using
- net defined by OOA [i] based on linear OOA(4256, 61683, F4, 34, 34) (dual of [(61683, 34), 2096966, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4256, 1048611, F4, 34) (dual of [1048611, 1048355, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(29) [i] based on
- linear OA(4251, 1048576, F4, 34) (dual of [1048576, 1048325, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4221, 1048576, F4, 30) (dual of [1048576, 1048355, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(33) ⊂ Ce(29) [i] based on
- OA 17-folding and stacking [i] based on linear OA(4256, 1048611, F4, 34) (dual of [1048611, 1048355, 35]-code), using
- net defined by OOA [i] based on linear OOA(4256, 61683, F4, 34, 34) (dual of [(61683, 34), 2096966, 35]-NRT-code), using
(259−34, 259, 405655)-Net over F4 — Digital
Digital (225, 259, 405655)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4259, 405655, F4, 2, 34) (dual of [(405655, 2), 811051, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4259, 524312, F4, 2, 34) (dual of [(524312, 2), 1048365, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4259, 1048624, F4, 34) (dual of [1048624, 1048365, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(4251, 1048576, F4, 34) (dual of [1048576, 1048325, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(48, 48, F4, 4) (dual of [48, 40, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(4259, 1048624, F4, 34) (dual of [1048624, 1048365, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(4259, 524312, F4, 2, 34) (dual of [(524312, 2), 1048365, 35]-NRT-code), using
(259−34, 259, large)-Net in Base 4 — Upper bound on s
There is no (225, 259, large)-net in base 4, because
- 32 times m-reduction [i] would yield (225, 227, large)-net in base 4, but