Best Known (113, 113+28, s)-Nets in Base 3
(113, 113+28, 688)-Net over F3 — Constructive and digital
Digital (113, 141, 688)-net over F3, using
- 31 times duplication [i] based on digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 172)-net over F81, using
(113, 113+28, 1932)-Net over F3 — Digital
Digital (113, 141, 1932)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 1932, F3, 28) (dual of [1932, 1791, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 2202, F3, 28) (dual of [2202, 2061, 29]-code), using
- (u, u+v)-construction [i] based on
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(3141, 2202, F3, 28) (dual of [2202, 2061, 29]-code), using
(113, 113+28, 193058)-Net in Base 3 — Upper bound on s
There is no (113, 141, 193059)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 18 797385 839098 520235 073828 824559 118734 299498 926844 236568 857224 511373 > 3141 [i]