Best Known (115, 115+16, s)-Nets in Base 4
(115, 115+16, 131086)-Net over F4 — Constructive and digital
Digital (115, 131, 131086)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (104, 120, 131072)-net over F4, using
- net defined by OOA [i] based on linear OOA(4120, 131072, F4, 16, 16) (dual of [(131072, 16), 2097032, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4120, 1048576, F4, 16) (dual of [1048576, 1048456, 17]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(4121, 1048577, F4, 17) (dual of [1048577, 1048456, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(4120, 1048576, F4, 16) (dual of [1048576, 1048456, 17]-code), using
- net defined by OOA [i] based on linear OOA(4120, 131072, F4, 16, 16) (dual of [(131072, 16), 2097032, 17]-NRT-code), using
- digital (3, 11, 14)-net over F4, using
(115, 115+16, 785033)-Net over F4 — Digital
Digital (115, 131, 785033)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4131, 785033, F4, 16) (dual of [785033, 784902, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4131, 1048636, F4, 16) (dual of [1048636, 1048505, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4131, 1048636, F4, 16) (dual of [1048636, 1048505, 17]-code), using
(115, 115+16, large)-Net in Base 4 — Upper bound on s
There is no (115, 131, large)-net in base 4, because
- 14 times m-reduction [i] would yield (115, 117, large)-net in base 4, but