Best Known (140−25, 140, s)-Nets in Base 4
(140−25, 140, 1374)-Net over F4 — Constructive and digital
Digital (115, 140, 1374)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 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 (102, 127, 1365)-net over F4, using
- net defined by OOA [i] based on linear OOA(4127, 1365, F4, 25, 25) (dual of [(1365, 25), 33998, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4127, 16381, F4, 25) (dual of [16381, 16254, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4127, 16381, F4, 25) (dual of [16381, 16254, 26]-code), using
- net defined by OOA [i] based on linear OOA(4127, 1365, F4, 25, 25) (dual of [(1365, 25), 33998, 26]-NRT-code), using
- digital (1, 13, 9)-net over F4, using
(140−25, 140, 13655)-Net over F4 — Digital
Digital (115, 140, 13655)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4140, 13655, F4, 25) (dual of [13655, 13515, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4140, 16432, F4, 25) (dual of [16432, 16292, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(413, 48, F4, 6) (dual of [48, 35, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4140, 16432, F4, 25) (dual of [16432, 16292, 26]-code), using
(140−25, 140, large)-Net in Base 4 — Upper bound on s
There is no (115, 140, large)-net in base 4, because
- 23 times m-reduction [i] would yield (115, 117, large)-net in base 4, but