Best Known (34, 49, s)-Nets in Base 4
(34, 49, 240)-Net over F4 — Constructive and digital
Digital (34, 49, 240)-net over F4, using
- 41 times duplication [i] based on digital (33, 48, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 16, 80)-net over F64, using
(34, 49, 284)-Net over F4 — Digital
Digital (34, 49, 284)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(449, 284, F4, 15) (dual of [284, 235, 16]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0) [i] based on linear OA(445, 263, F4, 15) (dual of [263, 218, 16]-code), using
- construction XX applied to C1 = C([254,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([254,13]) [i] based on
- linear OA(441, 255, F4, 14) (dual of [255, 214, 15]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(441, 255, F4, 14) (dual of [255, 214, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(445, 255, F4, 15) (dual of [255, 210, 16]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(437, 255, F4, 13) (dual of [255, 218, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(40, 4, F4, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(40, 4, F4, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([254,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([254,13]) [i] based on
- 17 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 9 times 0) [i] based on linear OA(445, 263, F4, 15) (dual of [263, 218, 16]-code), using
(34, 49, 15137)-Net in Base 4 — Upper bound on s
There is no (34, 49, 15138)-net in base 4, because
- 1 times m-reduction [i] would yield (34, 48, 15138)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 79244 379845 939024 912063 319760 > 448 [i]