Best Known (53, 74, s)-Nets in Base 3
(53, 74, 156)-Net over F3 — Constructive and digital
Digital (53, 74, 156)-net over F3, using
- 1 times m-reduction [i] based on digital (53, 75, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 25, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 25, 52)-net over F27, using
(53, 74, 255)-Net over F3 — Digital
Digital (53, 74, 255)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(374, 255, F3, 21) (dual of [255, 181, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(374, 260, F3, 21) (dual of [260, 186, 22]-code), using
- construction XX applied to C1 = C([102,121]), C2 = C([105,122]), C3 = C1 + C2 = C([105,121]), and C∩ = C1 ∩ C2 = C([102,122]) [i] based on
- linear OA(366, 242, F3, 20) (dual of [242, 176, 21]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {102,103,…,121}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(361, 242, F3, 18) (dual of [242, 181, 19]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {105,106,…,122}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(371, 242, F3, 21) (dual of [242, 171, 22]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {102,103,…,122}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {105,106,…,121}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(30, 5, F3, 0) (dual of [5, 5, 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
- construction XX applied to C1 = C([102,121]), C2 = C([105,122]), C3 = C1 + C2 = C([105,121]), and C∩ = C1 ∩ C2 = C([102,122]) [i] based on
- discarding factors / shortening the dual code based on linear OA(374, 260, F3, 21) (dual of [260, 186, 22]-code), using
(53, 74, 6875)-Net in Base 3 — Upper bound on s
There is no (53, 74, 6876)-net in base 3, because
- 1 times m-reduction [i] would yield (53, 73, 6876)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 67590 908346 474118 718458 505174 239049 > 373 [i]