Best Known (102−18, 102, s)-Nets in Base 4
(102−18, 102, 1830)-Net over F4 — Constructive and digital
Digital (84, 102, 1830)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-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 (1, 10, 9)-net over F4, using
(102−18, 102, 14307)-Net over F4 — Digital
Digital (84, 102, 14307)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4102, 14307, F4, 18) (dual of [14307, 14205, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4102, 16422, F4, 18) (dual of [16422, 16320, 19]-code), using
- 3 times code embedding in larger space [i] based on linear OA(499, 16419, F4, 18) (dual of [16419, 16320, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [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(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(499, 16419, F4, 18) (dual of [16419, 16320, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4102, 16422, F4, 18) (dual of [16422, 16320, 19]-code), using
(102−18, 102, large)-Net in Base 4 — Upper bound on s
There is no (84, 102, large)-net in base 4, because
- 16 times m-reduction [i] would yield (84, 86, large)-net in base 4, but