Best Known (174, 174+27, s)-Nets in Base 4
(174, 174+27, 80660)-Net over F4 — Constructive and digital
Digital (174, 201, 80660)-net over F4, using
- net defined by OOA [i] based on linear OOA(4201, 80660, F4, 27, 27) (dual of [(80660, 27), 2177619, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4201, 1048581, F4, 27) (dual of [1048581, 1048380, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 1048586, F4, 27) (dual of [1048586, 1048385, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 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(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(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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4201, 1048586, F4, 27) (dual of [1048586, 1048385, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4201, 1048581, F4, 27) (dual of [1048581, 1048380, 28]-code), using
(174, 174+27, 349528)-Net over F4 — Digital
Digital (174, 201, 349528)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4201, 349528, F4, 3, 27) (dual of [(349528, 3), 1048383, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4201, 1048584, F4, 27) (dual of [1048584, 1048383, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 1048586, F4, 27) (dual of [1048586, 1048385, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 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(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(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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4201, 1048586, F4, 27) (dual of [1048586, 1048385, 28]-code), using
- OOA 3-folding [i] based on linear OA(4201, 1048584, F4, 27) (dual of [1048584, 1048383, 28]-code), using
(174, 174+27, large)-Net in Base 4 — Upper bound on s
There is no (174, 201, large)-net in base 4, because
- 25 times m-reduction [i] would yield (174, 176, large)-net in base 4, but