Best Known (49, 49+23, s)-Nets in Base 4
(49, 49+23, 240)-Net over F4 — Constructive and digital
Digital (49, 72, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 24, 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
(49, 49+23, 296)-Net over F4 — Digital
Digital (49, 72, 296)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(472, 296, F4, 23) (dual of [296, 224, 24]-code), using
- 28 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 8 times 0, 1, 11 times 0) [i] based on linear OA(467, 263, F4, 23) (dual of [263, 196, 24]-code), using
- construction XX applied to C1 = C([254,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([254,21]) [i] based on
- linear OA(463, 255, F4, 22) (dual of [255, 192, 23]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(463, 255, F4, 22) (dual of [255, 192, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(467, 255, F4, 23) (dual of [255, 188, 24]-code), using the primitive BCH-code C(I) with length 255 = 44−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(459, 255, F4, 21) (dual of [255, 196, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [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([254,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([254,21]) [i] based on
- 28 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 8 times 0, 1, 11 times 0) [i] based on linear OA(467, 263, F4, 23) (dual of [263, 196, 24]-code), using
(49, 49+23, 12578)-Net in Base 4 — Upper bound on s
There is no (49, 72, 12579)-net in base 4, because
- 1 times m-reduction [i] would yield (49, 71, 12579)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5 579236 470292 585885 868639 962996 378836 096688 > 471 [i]