Best Known (47, 47+16, s)-Nets in Base 4
(47, 47+16, 514)-Net over F4 — Constructive and digital
Digital (47, 63, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(31,16) in PG(62,4)) for nets [i] based on digital (16, 32, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
(47, 47+16, 924)-Net over F4 — Digital
Digital (47, 63, 924)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(463, 924, F4, 16) (dual of [924, 861, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(463, 1037, F4, 16) (dual of [1037, 974, 17]-code), using
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(461, 1024, F4, 17) (dual of [1024, 963, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(451, 1024, F4, 14) (dual of [1024, 973, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(463, 1037, F4, 16) (dual of [1037, 974, 17]-code), using
(47, 47+16, 69143)-Net in Base 4 — Upper bound on s
There is no (47, 63, 69144)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 85 073706 162953 203215 941511 092710 498666 > 463 [i]