Best Known (35, 35+12, s)-Nets in Base 3
(35, 35+12, 164)-Net over F3 — Constructive and digital
Digital (35, 47, 164)-net over F3, using
- base reduction for projective spaces (embedding PG(23,9) in PG(46,3)) for nets [i] based on digital (12, 24, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 12, 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
- trace code for nets [i] based on digital (0, 12, 82)-net over F81, using
(35, 35+12, 291)-Net over F3 — Digital
Digital (35, 47, 291)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(347, 291, F3, 12) (dual of [291, 244, 13]-code), using
- 33 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0) [i] based on linear OA(341, 252, F3, 12) (dual of [252, 211, 13]-code), using
- construction XX applied to C1 = C([241,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([241,10]) [i] based on
- linear OA(336, 242, F3, 11) (dual of [242, 206, 12]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(336, 242, F3, 11) (dual of [242, 206, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(341, 242, F3, 12) (dual of [242, 201, 13]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(331, 242, F3, 10) (dual of [242, 211, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [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([241,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([241,10]) [i] based on
- 33 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0) [i] based on linear OA(341, 252, F3, 12) (dual of [252, 211, 13]-code), using
(35, 35+12, 8172)-Net in Base 3 — Upper bound on s
There is no (35, 47, 8173)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 26600 446757 118314 916177 > 347 [i]