Best Known (32, 43, s)-Nets in Base 4
(32, 43, 514)-Net over F4 — Constructive and digital
Digital (32, 43, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(21,16) in PG(42,4)) for nets [i] based on digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
(32, 43, 885)-Net over F4 — Digital
Digital (32, 43, 885)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(443, 885, F4, 11) (dual of [885, 842, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(443, 1040, F4, 11) (dual of [1040, 997, 12]-code), using
- construction XX applied to C1 = C([1021,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- linear OA(436, 1023, F4, 9) (dual of [1023, 987, 10]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(431, 1023, F4, 9) (dual of [1023, 992, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(441, 1023, F4, 11) (dual of [1023, 982, 12]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(426, 1023, F4, 7) (dual of [1023, 997, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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, 6, F4, 1) (dual of [6, 5, 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([1021,6]), C2 = C([0,8]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1021,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(443, 1040, F4, 11) (dual of [1040, 997, 12]-code), using
(32, 43, 99083)-Net in Base 4 — Upper bound on s
There is no (32, 43, 99084)-net in base 4, because
- 1 times m-reduction [i] would yield (32, 42, 99084)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 19 342964 501839 350088 676722 > 442 [i]