Best Known (44, 44+15, s)-Nets in Base 4
(44, 44+15, 514)-Net over F4 — Constructive and digital
Digital (44, 59, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(29,16) in PG(58,4)) for nets [i] based on digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
(44, 44+15, 908)-Net over F4 — Digital
Digital (44, 59, 908)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(459, 908, F4, 15) (dual of [908, 849, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(459, 1038, F4, 15) (dual of [1038, 979, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(456, 1024, F4, 15) (dual of [1024, 968, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(446, 1024, F4, 13) (dual of [1024, 978, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(441, 1024, F4, 11) (dual of [1024, 983, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 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, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(14) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(459, 1038, F4, 15) (dual of [1038, 979, 16]-code), using
(44, 44+15, 109716)-Net in Base 4 — Upper bound on s
There is no (44, 59, 109717)-net in base 4, because
- 1 times m-reduction [i] would yield (44, 58, 109717)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 83077 512851 557796 189779 850371 142800 > 458 [i]