Best Known (182, 221, s)-Nets in Base 3
(182, 221, 1480)-Net over F3 — Constructive and digital
Digital (182, 221, 1480)-net over F3, using
- 31 times duplication [i] based on digital (181, 220, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 55, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 55, 370)-net over F81, using
(182, 221, 4998)-Net over F3 — Digital
Digital (182, 221, 4998)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 4998, F3, 39) (dual of [4998, 4777, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 6606, F3, 39) (dual of [6606, 6385, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(312, 44, F3, 5) (dual of [44, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3221, 6606, F3, 39) (dual of [6606, 6385, 40]-code), using
(182, 221, 1326604)-Net in Base 3 — Upper bound on s
There is no (182, 221, 1326605)-net in base 3, because
- 1 times m-reduction [i] would yield (182, 220, 1326605)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 926 150004 677983 075482 054426 920534 725735 670941 396774 969889 545477 585282 002425 358809 342752 846065 768534 931467 > 3220 [i]