Best Known (24, 24+11, s)-Nets in Base 4
(24, 24+11, 195)-Net over F4 — Constructive and digital
Digital (24, 35, 195)-net over F4, using
- 1 times m-reduction [i] based on digital (24, 36, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 12, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 12, 65)-net over F64, using
(24, 24+11, 254)-Net over F4 — Digital
Digital (24, 35, 254)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(435, 254, F4, 11) (dual of [254, 219, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(435, 269, F4, 11) (dual of [269, 234, 12]-code), using
- construction XX applied to C1 = C([253,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- linear OA(429, 255, F4, 9) (dual of [255, 226, 10]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(425, 255, F4, 9) (dual of [255, 230, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(433, 255, F4, 11) (dual of [255, 222, 12]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(421, 255, F4, 7) (dual of [255, 234, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(41, 5, F4, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to C1 = C([253,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(435, 269, F4, 11) (dual of [269, 234, 12]-code), using
(24, 24+11, 10778)-Net in Base 4 — Upper bound on s
There is no (24, 35, 10779)-net in base 4, because
- 1 times m-reduction [i] would yield (24, 34, 10779)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 295 158860 125397 396884 > 434 [i]