Best Known (33, 33+11, s)-Nets in Base 4
(33, 33+11, 1028)-Net over F4 — Constructive and digital
Digital (33, 44, 1028)-net over F4, 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
(33, 33+11, 1034)-Net over F4 — Digital
Digital (33, 44, 1034)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(444, 1034, F4, 11) (dual of [1034, 990, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(444, 1041, F4, 11) (dual of [1041, 997, 12]-code), using
- construction XX applied to C1 = C([332,341]), C2 = C([335,342]), C3 = C1 + C2 = C([335,341]), and C∩ = C1 ∩ C2 = C([332,342]) [i] based on
- linear OA(436, 1023, F4, 10) (dual of [1023, 987, 11]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {332,333,…,341}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(431, 1023, F4, 8) (dual of [1023, 992, 9]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {335,336,…,342}, and designed minimum distance d ≥ |I|+1 = 9 [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 = {332,333,…,342}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(426, 1023, F4, 7) (dual of [1023, 997, 8]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {335,336,…,341}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(43, 13, F4, 2) (dual of [13, 10, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- 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
- construction XX applied to C1 = C([332,341]), C2 = C([335,342]), C3 = C1 + C2 = C([335,341]), and C∩ = C1 ∩ C2 = C([332,342]) [i] based on
- discarding factors / shortening the dual code based on linear OA(444, 1041, F4, 11) (dual of [1041, 997, 12]-code), using
(33, 33+11, 130743)-Net in Base 4 — Upper bound on s
There is no (33, 44, 130744)-net in base 4, because
- 1 times m-reduction [i] would yield (33, 43, 130744)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 77 373948 955642 346988 145303 > 443 [i]