Best Known (243−51, 243, s)-Nets in Base 3
(243−51, 243, 688)-Net over F3 — Constructive and digital
Digital (192, 243, 688)-net over F3, using
- t-expansion [i] based on digital (190, 243, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (190, 244, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 61, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 61, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (190, 244, 688)-net over F3, using
(243−51, 243, 2126)-Net over F3 — Digital
Digital (192, 243, 2126)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3243, 2126, F3, 51) (dual of [2126, 1883, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(3243, 2207, F3, 51) (dual of [2207, 1964, 52]-code), using
- construction XX applied to Ce(51) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using an extension Ce(51) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,51], and designed minimum distance d ≥ |I|+1 = 52 [i]
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(51) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- discarding factors / shortening the dual code based on linear OA(3243, 2207, F3, 51) (dual of [2207, 1964, 52]-code), using
(243−51, 243, 211387)-Net in Base 3 — Upper bound on s
There is no (192, 243, 211388)-net in base 3, because
- 1 times m-reduction [i] would yield (192, 242, 211388)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 29 065024 856831 855833 487546 879351 978868 624597 762202 455984 997475 206778 976978 685358 284986 384826 469837 363348 971256 066553 > 3242 [i]