Best Known (128−16, 128, s)-Nets in Base 4
(128−16, 128, 131077)-Net over F4 — Constructive and digital
Digital (112, 128, 131077)-net over F4, using
- net defined by OOA [i] based on linear OOA(4128, 131077, F4, 16, 16) (dual of [(131077, 16), 2097104, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4128, 1048616, F4, 16) (dual of [1048616, 1048488, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4128, 1048620, F4, 16) (dual of [1048620, 1048492, 17]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(481, 1048577, F4, 11) (dual of [1048577, 1048496, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4128, 1048620, F4, 16) (dual of [1048620, 1048492, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(4128, 1048616, F4, 16) (dual of [1048616, 1048488, 17]-code), using
(128−16, 128, 583274)-Net over F4 — Digital
Digital (112, 128, 583274)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4128, 583274, F4, 16) (dual of [583274, 583146, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4128, 1048620, F4, 16) (dual of [1048620, 1048492, 17]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(481, 1048577, F4, 11) (dual of [1048577, 1048496, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4128, 1048620, F4, 16) (dual of [1048620, 1048492, 17]-code), using
(128−16, 128, large)-Net in Base 4 — Upper bound on s
There is no (112, 128, large)-net in base 4, because
- 14 times m-reduction [i] would yield (112, 114, large)-net in base 4, but