Best Known (257−57, 257, s)-Nets in Base 4
(257−57, 257, 1539)-Net over F4 — Constructive and digital
Digital (200, 257, 1539)-net over F4, using
- 1 times m-reduction [i] based on digital (200, 258, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 86, 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, 86, 513)-net over F64, using
(257−57, 257, 4234)-Net over F4 — Digital
Digital (200, 257, 4234)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4257, 4234, F4, 57) (dual of [4234, 3977, 58]-code), using
- 133 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 14 times 0, 1, 40 times 0, 1, 72 times 0) [i] based on linear OA(4253, 4097, F4, 57) (dual of [4097, 3844, 58]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,28], and minimum distance d ≥ |{−28,−27,…,28}|+1 = 58 (BCH-bound) [i]
- 133 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 14 times 0, 1, 40 times 0, 1, 72 times 0) [i] based on linear OA(4253, 4097, F4, 57) (dual of [4097, 3844, 58]-code), using
(257−57, 257, 1203438)-Net in Base 4 — Upper bound on s
There is no (200, 257, 1203439)-net in base 4, because
- 1 times m-reduction [i] would yield (200, 256, 1203439)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 13407 855245 727781 238552 391776 864572 010386 907583 315782 343219 478046 093810 000125 364695 959078 752109 851495 018474 688805 727753 426287 463326 861288 982162 237548 931650 > 4256 [i]