Best Known (182, 182+48, s)-Nets in Base 3
(182, 182+48, 688)-Net over F3 — Constructive and digital
Digital (182, 230, 688)-net over F3, using
- t-expansion [i] based on digital (181, 230, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 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, 58, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
(182, 182+48, 2093)-Net over F3 — Digital
Digital (182, 230, 2093)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3230, 2093, F3, 48) (dual of [2093, 1863, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(3230, 2209, F3, 48) (dual of [2209, 1979, 49]-code), using
- construction XX applied to Ce(48) ⊂ Ce(45) ⊂ Ce(43) [i] based on
- 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(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(48) ⊂ Ce(45) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3230, 2209, F3, 48) (dual of [2209, 1979, 49]-code), using
(182, 182+48, 183100)-Net in Base 3 — Upper bound on s
There is no (182, 230, 183101)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 54 692654 187282 671095 667790 133787 851819 828164 662925 248987 098625 519581 562025 975797 825024 446101 204095 477053 556001 > 3230 [i]