Best Known (260−57, 260, s)-Nets in Base 4
(260−57, 260, 1539)-Net over F4 — Constructive and digital
Digital (203, 260, 1539)-net over F4, using
- 42 times duplication [i] based on digital (201, 258, 1539)-net over F4, using
- t-expansion [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
- t-expansion [i] based on digital (200, 258, 1539)-net over F4, using
(260−57, 260, 4545)-Net over F4 — Digital
Digital (203, 260, 4545)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4260, 4545, F4, 57) (dual of [4545, 4285, 58]-code), using
- 441 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 14 times 0, 1, 40 times 0, 1, 72 times 0, 1, 94 times 0, 1, 103 times 0, 1, 108 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]
- 441 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 14 times 0, 1, 40 times 0, 1, 72 times 0, 1, 94 times 0, 1, 103 times 0, 1, 108 times 0) [i] based on linear OA(4253, 4097, F4, 57) (dual of [4097, 3844, 58]-code), using
(260−57, 260, 1396148)-Net in Base 4 — Upper bound on s
There is no (203, 260, 1396149)-net in base 4, because
- 1 times m-reduction [i] would yield (203, 259, 1396149)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 858109 540423 940115 058205 999769 192554 572239 716179 393052 948561 219423 014815 629149 122941 464988 099750 127498 481304 737211 948676 501840 420709 168775 160153 187101 820468 > 4259 [i]