Best Known (147, 147+31, s)-Nets in Base 4
(147, 147+31, 1539)-Net over F4 — Constructive and digital
Digital (147, 178, 1539)-net over F4, using
- 41 times duplication [i] based on digital (146, 177, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 59, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 59, 513)-net over F64, using
(147, 147+31, 16444)-Net over F4 — Digital
Digital (147, 178, 16444)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4178, 16444, F4, 31) (dual of [16444, 16266, 32]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4176, 16440, F4, 31) (dual of [16440, 16264, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(4176, 16442, F4, 30) (dual of [16442, 16266, 31]-code), using Gilbert–Varšamov bound and bm = 4176 > Vbs−1(k−1) = 1384 909020 997011 203617 739567 249150 007638 670237 712683 024848 402876 238876 604983 610276 576069 059800 658653 690320 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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(4176, 16440, F4, 31) (dual of [16440, 16264, 32]-code), using
- construction X with Varšamov bound [i] based on
(147, 147+31, large)-Net in Base 4 — Upper bound on s
There is no (147, 178, large)-net in base 4, because
- 29 times m-reduction [i] would yield (147, 149, large)-net in base 4, but