Best Known (173, 220, s)-Nets in Base 3
(173, 220, 688)-Net over F3 — Constructive and digital
Digital (173, 220, 688)-net over F3, using
- t-expansion [i] based on digital (172, 220, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 55, 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, 55, 172)-net over F81, using
(173, 220, 1808)-Net over F3 — Digital
Digital (173, 220, 1808)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3220, 1808, F3, 47) (dual of [1808, 1588, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 2197, F3, 47) (dual of [2197, 1977, 48]-code), using
- construction XX applied to Ce(46) ⊂ Ce(45) ⊂ Ce(43) [i] based on
- 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(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(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(46) ⊂ Ce(45) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3220, 2197, F3, 47) (dual of [2197, 1977, 48]-code), using
(173, 220, 164585)-Net in Base 3 — Upper bound on s
There is no (173, 220, 164586)-net in base 3, because
- 1 times m-reduction [i] would yield (173, 219, 164586)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 308 743138 227477 145518 178040 159038 574861 891674 298647 484037 641933 958125 472762 535763 275843 893503 281720 076073 > 3219 [i]