Best Known (213−29, 213, s)-Nets in Base 4
(213−29, 213, 74899)-Net over F4 — Constructive and digital
Digital (184, 213, 74899)-net over F4, using
- 41 times duplication [i] based on digital (183, 212, 74899)-net over F4, using
- net defined by OOA [i] based on linear OOA(4212, 74899, F4, 29, 29) (dual of [(74899, 29), 2171859, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4212, 1048587, F4, 29) (dual of [1048587, 1048375, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 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(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(28) ⊂ Ce(26) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(4212, 1048587, F4, 29) (dual of [1048587, 1048375, 30]-code), using
- net defined by OOA [i] based on linear OOA(4212, 74899, F4, 29, 29) (dual of [(74899, 29), 2171859, 30]-NRT-code), using
(213−29, 213, 349529)-Net over F4 — Digital
Digital (184, 213, 349529)-net over F4, using
- 41 times duplication [i] based on digital (183, 212, 349529)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4212, 349529, F4, 3, 29) (dual of [(349529, 3), 1048375, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4212, 1048587, F4, 29) (dual of [1048587, 1048375, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 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(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(28) ⊂ Ce(26) [i] based on
- OOA 3-folding [i] based on linear OA(4212, 1048587, F4, 29) (dual of [1048587, 1048375, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4212, 349529, F4, 3, 29) (dual of [(349529, 3), 1048375, 30]-NRT-code), using
(213−29, 213, large)-Net in Base 4 — Upper bound on s
There is no (184, 213, large)-net in base 4, because
- 27 times m-reduction [i] would yield (184, 186, large)-net in base 4, but