Best Known (25−7, 25, s)-Nets in Base 4
(25−7, 25, 257)-Net over F4 — Constructive and digital
Digital (18, 25, 257)-net over F4, using
- base reduction for projective spaces (embedding PG(6,256) in PG(24,4)) for nets [i] based on digital (0, 7, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
(25−7, 25, 351)-Net over F4 — Digital
Digital (18, 25, 351)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(425, 351, F4, 7) (dual of [351, 326, 8]-code), using
- construction XX applied to C1 = C([111,116]), C2 = C([110,115]), C3 = C1 + C2 = C([111,115]), and C∩ = C1 ∩ C2 = C([110,116]) [i] based on
- linear OA(420, 341, F4, 6) (dual of [341, 321, 7]-code), using the BCH-code C(I) with length 341 | 45−1, defining interval I = {111,112,…,116}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(420, 341, F4, 6) (dual of [341, 321, 7]-code), using the BCH-code C(I) with length 341 | 45−1, defining interval I = {110,111,…,115}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 341, F4, 7) (dual of [341, 316, 8]-code), using the BCH-code C(I) with length 341 | 45−1, defining interval I = {110,111,…,116}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(415, 341, F4, 5) (dual of [341, 326, 6]-code), using the BCH-code C(I) with length 341 | 45−1, defining interval I = {111,112,113,114,115}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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
- linear OA(40, 5, F4, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([111,116]), C2 = C([110,115]), C3 = C1 + C2 = C([111,115]), and C∩ = C1 ∩ C2 = C([110,116]) [i] based on
(25−7, 25, 39693)-Net in Base 4 — Upper bound on s
There is no (18, 25, 39694)-net in base 4, because
- 1 times m-reduction [i] would yield (18, 24, 39694)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 281 483377 321762 > 424 [i]