Best Known (178, 178+33, s)-Nets in Base 3
(178, 178+33, 1480)-Net over F3 — Constructive and digital
Digital (178, 211, 1480)-net over F3, using
- 5 times m-reduction [i] based on digital (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
(178, 178+33, 10566)-Net over F3 — Digital
Digital (178, 211, 10566)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3211, 10566, F3, 33) (dual of [10566, 10355, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 19732, F3, 33) (dual of [19732, 19521, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(3199, 19684, F3, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3163, 19684, F3, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(312, 48, F3, 5) (dual of [48, 36, 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([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3211, 19732, F3, 33) (dual of [19732, 19521, 34]-code), using
(178, 178+33, 6219050)-Net in Base 3 — Upper bound on s
There is no (178, 211, 6219051)-net in base 3, because
- 1 times m-reduction [i] would yield (178, 210, 6219051)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 15684 242493 098385 395307 733541 771841 164785 254372 724793 907844 120838 869853 493111 822879 990513 050972 250081 > 3210 [i]