Best Known (233−51, 233, s)-Nets in Base 4
(233−51, 233, 1056)-Net over F4 — Constructive and digital
Digital (182, 233, 1056)-net over F4, using
- 41 times duplication [i] based on digital (181, 232, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 58, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
- trace code for nets [i] based on digital (7, 58, 264)-net over F256, using
(233−51, 233, 4206)-Net over F4 — Digital
Digital (182, 233, 4206)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4233, 4206, F4, 51) (dual of [4206, 3973, 52]-code), using
- 93 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0, 1, 24 times 0, 1, 59 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
- 93 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0, 1, 24 times 0, 1, 59 times 0) [i] based on linear OA(4230, 4110, F4, 51) (dual of [4110, 3880, 52]-code), using
(233−51, 233, 1311035)-Net in Base 4 — Upper bound on s
There is no (182, 233, 1311036)-net in base 4, because
- 1 times m-reduction [i] would yield (182, 232, 1311036)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 47 634269 390183 114768 653520 248923 504582 027022 097883 553592 116433 160335 649073 075961 613751 930029 945157 861561 783142 580666 413633 935781 427370 432987 > 4232 [i]