Best Known (75−35, 75, s)-Nets in Base 27
(75−35, 75, 192)-Net over F27 — Constructive and digital
Digital (40, 75, 192)-net over F27, using
- 1 times m-reduction [i] based on digital (40, 76, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 29, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (11, 47, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 29, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(75−35, 75, 370)-Net in Base 27 — Constructive
(40, 75, 370)-net in base 27, using
- 21 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 72, 370)-net over F81, using
(75−35, 75, 803)-Net over F27 — Digital
Digital (40, 75, 803)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2775, 803, F27, 35) (dual of [803, 728, 36]-code), using
- 62 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 29 times 0) [i] based on linear OA(2766, 732, F27, 35) (dual of [732, 666, 36]-code), using
- construction XX applied to C1 = C([727,32]), C2 = C([0,33]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([727,33]) [i] based on
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,32}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2764, 728, F27, 34) (dual of [728, 664, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2766, 728, F27, 35) (dual of [728, 662, 36]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,33}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2762, 728, F27, 33) (dual of [728, 666, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,32]), C2 = C([0,33]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([727,33]) [i] based on
- 62 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 8 times 0, 1, 16 times 0, 1, 29 times 0) [i] based on linear OA(2766, 732, F27, 35) (dual of [732, 666, 36]-code), using
(75−35, 75, 469468)-Net in Base 27 — Upper bound on s
There is no (40, 75, 469469)-net in base 27, because
- 1 times m-reduction [i] would yield (40, 74, 469469)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 8335 509191 172267 604063 848861 524066 070486 318744 023446 545670 751550 220154 791476 057709 262349 188871 801737 563827 > 2774 [i]