Best Known (247−39, 247, s)-Nets in Base 3
(247−39, 247, 1496)-Net over F3 — Constructive and digital
Digital (208, 247, 1496)-net over F3, using
- 31 times duplication [i] based on digital (207, 246, 1496)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (7, 26, 16)-net over F3, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 7 and N(F) ≥ 16, using
- net from sequence [i] based on digital (7, 15)-sequence over F3, using
- 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
- digital (7, 26, 16)-net over F3, using
- (u, u+v)-construction [i] based on
(247−39, 247, 10856)-Net over F3 — Digital
Digital (208, 247, 10856)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3247, 10856, F3, 39) (dual of [10856, 10609, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3247, 19732, F3, 39) (dual of [19732, 19485, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3235, 19684, F3, 39) (dual of [19684, 19449, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- 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(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([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3247, 19732, F3, 39) (dual of [19732, 19485, 40]-code), using
(247−39, 247, 5965527)-Net in Base 3 — Upper bound on s
There is no (208, 247, 5965528)-net in base 3, because
- 1 times m-reduction [i] would yield (208, 246, 5965528)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2354 126339 958010 653318 912521 313160 727710 730011 652373 971020 836799 124027 228566 775632 132887 339514 858973 052327 912763 404513 > 3246 [i]