Best Known (231−51, 231, s)-Nets in Base 4
(231−51, 231, 1052)-Net over F4 — Constructive and digital
Digital (180, 231, 1052)-net over F4, using
- 1 times m-reduction [i] based on digital (180, 232, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 58, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 58, 263)-net over F256, using
(231−51, 231, 4119)-Net over F4 — Digital
Digital (180, 231, 4119)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4231, 4119, F4, 51) (dual of [4119, 3888, 52]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(4230, 4110, F4, 51) (dual of [4110, 3880, 52]-code), using
- construction X applied to C([0,25]) ⊂ C([0,24]) [i] based on
- linear OA(4229, 4097, F4, 51) (dual of [4097, 3868, 52]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,25], and minimum distance d ≥ |{−25,−24,…,25}|+1 = 52 (BCH-bound) [i]
- linear OA(4217, 4097, F4, 49) (dual of [4097, 3880, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to C([0,25]) ⊂ C([0,24]) [i] based on
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(4230, 4110, F4, 51) (dual of [4110, 3880, 52]-code), using
(231−51, 231, 1173407)-Net in Base 4 — Upper bound on s
There is no (180, 231, 1173408)-net in base 4, because
- 1 times m-reduction [i] would yield (180, 230, 1173408)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 977144 727574 110578 857759 426286 910026 909359 437078 117141 067806 970871 384862 696228 412589 693236 327445 020216 662077 684961 278656 230749 384647 070927 > 4230 [i]