Best Known (119−16, 119, s)-Nets in Base 4
(119−16, 119, 32782)-Net over F4 — Constructive and digital
Digital (103, 119, 32782)-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 (92, 108, 32768)-net over F4, using
- net defined by OOA [i] based on linear OOA(4108, 32768, F4, 16, 16) (dual of [(32768, 16), 524180, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4108, 262144, F4, 16) (dual of [262144, 262036, 17]-code), using
- 1 times truncation [i] based on linear OA(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, 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(4109, 262145, F4, 17) (dual of [262145, 262036, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(4108, 262144, F4, 16) (dual of [262144, 262036, 17]-code), using
- net defined by OOA [i] based on linear OOA(4108, 32768, F4, 16, 16) (dual of [(32768, 16), 524180, 17]-NRT-code), using
- digital (3, 11, 14)-net over F4, using
(119−16, 119, 239234)-Net over F4 — Digital
Digital (103, 119, 239234)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4119, 239234, F4, 16) (dual of [239234, 239115, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4119, 262199, F4, 16) (dual of [262199, 262080, 17]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 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(4119, 262199, F4, 16) (dual of [262199, 262080, 17]-code), using
(119−16, 119, large)-Net in Base 4 — Upper bound on s
There is no (103, 119, large)-net in base 4, because
- 14 times m-reduction [i] would yield (103, 105, large)-net in base 4, but