Best Known (178, 178+27, s)-Nets in Base 4
(178, 178+27, 80661)-Net over F4 — Constructive and digital
Digital (178, 205, 80661)-net over F4, using
- 43 times duplication [i] based on digital (175, 202, 80661)-net over F4, using
- net defined by OOA [i] based on linear OOA(4202, 80661, F4, 27, 27) (dual of [(80661, 27), 2177645, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4202, 1048594, F4, 27) (dual of [1048594, 1048392, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4202, 1048598, F4, 27) (dual of [1048598, 1048396, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4201, 1048577, F4, 27) (dual of [1048577, 1048376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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 C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4202, 1048598, F4, 27) (dual of [1048598, 1048396, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4202, 1048594, F4, 27) (dual of [1048594, 1048392, 28]-code), using
- net defined by OOA [i] based on linear OOA(4202, 80661, F4, 27, 27) (dual of [(80661, 27), 2177645, 28]-NRT-code), using
(178, 178+27, 404245)-Net over F4 — Digital
Digital (178, 205, 404245)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4205, 404245, F4, 2, 27) (dual of [(404245, 2), 808285, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4205, 524300, F4, 2, 27) (dual of [(524300, 2), 1048395, 28]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4204, 524300, F4, 2, 27) (dual of [(524300, 2), 1048396, 28]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4202, 524299, F4, 2, 27) (dual of [(524299, 2), 1048396, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4202, 1048598, F4, 27) (dual of [1048598, 1048396, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4201, 1048577, F4, 27) (dual of [1048577, 1048376, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 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 C([0,13]) ⊂ C([0,12]) [i] based on
- OOA 2-folding [i] based on linear OA(4202, 1048598, F4, 27) (dual of [1048598, 1048396, 28]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4202, 524299, F4, 2, 27) (dual of [(524299, 2), 1048396, 28]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4204, 524300, F4, 2, 27) (dual of [(524300, 2), 1048396, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4205, 524300, F4, 2, 27) (dual of [(524300, 2), 1048395, 28]-NRT-code), using
(178, 178+27, large)-Net in Base 4 — Upper bound on s
There is no (178, 205, large)-net in base 4, because
- 25 times m-reduction [i] would yield (178, 180, large)-net in base 4, but