Best Known (202−27, 202, s)-Nets in Base 4
(202−27, 202, 80661)-Net over F4 — Constructive and digital
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
(202−27, 202, 349532)-Net over F4 — Digital
Digital (175, 202, 349532)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4202, 349532, F4, 3, 27) (dual of [(349532, 3), 1048394, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4202, 1048596, F4, 27) (dual of [1048596, 1048394, 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 3-folding [i] based on linear OA(4202, 1048596, F4, 27) (dual of [1048596, 1048394, 28]-code), using
(202−27, 202, large)-Net in Base 4 — Upper bound on s
There is no (175, 202, large)-net in base 4, because
- 25 times m-reduction [i] would yield (175, 177, large)-net in base 4, but