Best Known (165, 165+26, s)-Nets in Base 4
(165, 165+26, 80660)-Net over F4 — Constructive and digital
Digital (165, 191, 80660)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 80660, F4, 26, 26) (dual of [(80660, 26), 2096969, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4191, 1048580, F4, 26) (dual of [1048580, 1048389, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(40, 10, F4, 0) (dual of [10, 10, 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(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4191, 1048580, F4, 26) (dual of [1048580, 1048389, 27]-code), using
(165, 165+26, 349528)-Net over F4 — Digital
Digital (165, 191, 349528)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4191, 349528, F4, 3, 26) (dual of [(349528, 3), 1048393, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4191, 1048584, F4, 26) (dual of [1048584, 1048393, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(40, 10, F4, 0) (dual of [10, 10, 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(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- OOA 3-folding [i] based on linear OA(4191, 1048584, F4, 26) (dual of [1048584, 1048393, 27]-code), using
(165, 165+26, large)-Net in Base 4 — Upper bound on s
There is no (165, 191, large)-net in base 4, because
- 24 times m-reduction [i] would yield (165, 167, large)-net in base 4, but