Best Known (101−18, 101, s)-Nets in Base 4
(101−18, 101, 1826)-Net over F4 — Constructive and digital
Digital (83, 101, 1826)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (74, 92, 1821)-net over F4, using
- net defined by OOA [i] based on linear OOA(492, 1821, F4, 18, 18) (dual of [(1821, 18), 32686, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(492, 16389, F4, 18) (dual of [16389, 16297, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(492, 16391, F4, 18) (dual of [16391, 16299, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(492, 16391, F4, 18) (dual of [16391, 16299, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(492, 16389, F4, 18) (dual of [16389, 16297, 19]-code), using
- net defined by OOA [i] based on linear OOA(492, 1821, F4, 18, 18) (dual of [(1821, 18), 32686, 19]-NRT-code), using
- digital (0, 9, 5)-net over F4, using
(101−18, 101, 13119)-Net over F4 — Digital
Digital (83, 101, 13119)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4101, 13119, F4, 18) (dual of [13119, 13018, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4101, 16394, F4, 18) (dual of [16394, 16293, 19]-code), using
- (u, u+v)-construction [i] based on
- linear OA(49, 10, F4, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,4)), using
- dual of repetition code with length 10 [i]
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(49, 10, F4, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,4)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4101, 16394, F4, 18) (dual of [16394, 16293, 19]-code), using
(101−18, 101, 7890069)-Net in Base 4 — Upper bound on s
There is no (83, 101, 7890070)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6 427756 756742 354237 095462 599837 882903 163058 402091 021129 584758 > 4101 [i]