Best Known (108, 108+23, s)-Nets in Base 4
(108, 108+23, 1495)-Net over F4 — Constructive and digital
Digital (108, 131, 1495)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (97, 120, 1490)-net over F4, using
- net defined by OOA [i] based on linear OOA(4120, 1490, F4, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4120, 16391, F4, 23) (dual of [16391, 16271, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(4120, 16391, F4, 23) (dual of [16391, 16271, 24]-code), using
- net defined by OOA [i] based on linear OOA(4120, 1490, F4, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
- digital (0, 11, 5)-net over F4, using
(108, 108+23, 15415)-Net over F4 — Digital
Digital (108, 131, 15415)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4131, 15415, F4, 23) (dual of [15415, 15284, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4131, 16396, F4, 23) (dual of [16396, 16265, 24]-code), using
- (u, u+v)-construction [i] based on
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4131, 16396, F4, 23) (dual of [16396, 16265, 24]-code), using
(108, 108+23, large)-Net in Base 4 — Upper bound on s
There is no (108, 131, large)-net in base 4, because
- 21 times m-reduction [i] would yield (108, 110, large)-net in base 4, but