Best Known (45−11, 45, s)-Nets in Base 4
(45−11, 45, 1028)-Net over F4 — Constructive and digital
Digital (34, 45, 1028)-net over F4, using
- 41 times duplication [i] based on 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
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
(45−11, 45, 1063)-Net over F4 — Digital
Digital (34, 45, 1063)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(445, 1063, F4, 11) (dual of [1063, 1018, 12]-code), using
- 26 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 15 times 0) [i] based on linear OA(441, 1033, F4, 11) (dual of [1033, 992, 12]-code), using
- construction XX applied to C1 = C([1022,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1022,9]) [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 = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(436, 1023, F4, 10) (dual of [1023, 987, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [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 = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [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(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([1022,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([1022,9]) [i] based on
- 26 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 15 times 0) [i] based on linear OA(441, 1033, F4, 11) (dual of [1033, 992, 12]-code), using
(45−11, 45, 172517)-Net in Base 4 — Upper bound on s
There is no (34, 45, 172518)-net in base 4, because
- 1 times m-reduction [i] would yield (34, 44, 172518)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 309 487599 454998 436782 409423 > 444 [i]