Best Known (122−17, 122, s)-Nets in Base 4
(122−17, 122, 131073)-Net over F4 — Constructive and digital
Digital (105, 122, 131073)-net over F4, using
- net defined by OOA [i] based on linear OOA(4122, 131073, F4, 17, 17) (dual of [(131073, 17), 2228119, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4122, 1048585, F4, 17) (dual of [1048585, 1048463, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4122, 1048587, F4, 17) (dual of [1048587, 1048465, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4122, 1048587, F4, 17) (dual of [1048587, 1048465, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4122, 1048585, F4, 17) (dual of [1048585, 1048463, 18]-code), using
(122−17, 122, 349529)-Net over F4 — Digital
Digital (105, 122, 349529)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4122, 349529, F4, 3, 17) (dual of [(349529, 3), 1048465, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4122, 1048587, F4, 17) (dual of [1048587, 1048465, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(16) ⊂ Ce(14) [i] based on
- OOA 3-folding [i] based on linear OA(4122, 1048587, F4, 17) (dual of [1048587, 1048465, 18]-code), using
(122−17, 122, large)-Net in Base 4 — Upper bound on s
There is no (105, 122, large)-net in base 4, because
- 15 times m-reduction [i] would yield (105, 107, large)-net in base 4, but