Best Known (25, 34, s)-Nets in Base 3
(25, 34, 164)-Net over F3 — Constructive and digital
Digital (25, 34, 164)-net over F3, using
- trace code for nets [i] based on digital (8, 17, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(8,81) in PG(16,9)) for nets [i] based on digital (0, 9, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base reduction for projective spaces (embedding PG(8,81) in PG(16,9)) for nets [i] based on digital (0, 9, 82)-net over F81, using
(25, 34, 263)-Net over F3 — Digital
Digital (25, 34, 263)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(334, 263, F3, 9) (dual of [263, 229, 10]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0) [i] based on linear OA(331, 252, F3, 9) (dual of [252, 221, 10]-code), using
- construction XX applied to C1 = C([241,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([241,7]) [i] based on
- linear OA(326, 242, F3, 8) (dual of [242, 216, 9]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(326, 242, F3, 8) (dual of [242, 216, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(331, 242, F3, 9) (dual of [242, 211, 10]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(321, 242, F3, 7) (dual of [242, 221, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [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
- linear OA(30, 5, F3, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([241,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([241,7]) [i] based on
- 8 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0) [i] based on linear OA(331, 252, F3, 9) (dual of [252, 221, 10]-code), using
(25, 34, 9552)-Net in Base 3 — Upper bound on s
There is no (25, 34, 9553)-net in base 3, because
- 1 times m-reduction [i] would yield (25, 33, 9553)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5560 375230 238929 > 333 [i]