Best Known (98−39, 98, s)-Nets in Base 27
(98−39, 98, 270)-Net over F27 — Constructive and digital
Digital (59, 98, 270)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (7, 20, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (10, 29, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (10, 49, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27 (see above)
- digital (7, 20, 82)-net over F27, using
(98−39, 98, 438)-Net in Base 27 — Constructive
(59, 98, 438)-net in base 27, using
- (u, u+v)-construction [i] based on
- digital (5, 24, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- (35, 74, 370)-net in base 27, using
- 2 times m-reduction [i] based on (35, 76, 370)-net in base 27, using
- base change [i] based on digital (16, 57, 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, 57, 370)-net over F81, using
- 2 times m-reduction [i] based on (35, 76, 370)-net in base 27, using
- digital (5, 24, 68)-net over F27, using
(98−39, 98, 2859)-Net over F27 — Digital
Digital (59, 98, 2859)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2798, 2859, F27, 39) (dual of [2859, 2761, 40]-code), using
- 2760 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 30 times 0, 1, 33 times 0, 1, 37 times 0, 1, 39 times 0, 1, 44 times 0, 1, 47 times 0, 1, 52 times 0, 1, 57 times 0, 1, 62 times 0, 1, 68 times 0, 1, 75 times 0, 1, 81 times 0, 1, 89 times 0, 1, 97 times 0, 1, 106 times 0, 1, 116 times 0, 1, 126 times 0, 1, 138 times 0, 1, 151 times 0, 1, 165 times 0, 1, 180 times 0, 1, 196 times 0, 1, 215 times 0, 1, 234 times 0) [i] based on linear OA(2739, 40, F27, 39) (dual of [40, 1, 40]-code or 40-arc in PG(38,27)), using
- dual of repetition code with length 40 [i]
- 2760 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 30 times 0, 1, 33 times 0, 1, 37 times 0, 1, 39 times 0, 1, 44 times 0, 1, 47 times 0, 1, 52 times 0, 1, 57 times 0, 1, 62 times 0, 1, 68 times 0, 1, 75 times 0, 1, 81 times 0, 1, 89 times 0, 1, 97 times 0, 1, 106 times 0, 1, 116 times 0, 1, 126 times 0, 1, 138 times 0, 1, 151 times 0, 1, 165 times 0, 1, 180 times 0, 1, 196 times 0, 1, 215 times 0, 1, 234 times 0) [i] based on linear OA(2739, 40, F27, 39) (dual of [40, 1, 40]-code or 40-arc in PG(38,27)), using
(98−39, 98, 6190563)-Net in Base 27 — Upper bound on s
There is no (59, 98, 6190564)-net in base 27, because
- 1 times m-reduction [i] would yield (59, 97, 6190564)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 6 954823 283947 121718 108411 755395 379918 395313 776419 416573 000913 324149 830901 760688 195103 792865 117938 200596 466623 632901 779950 999718 782544 961681 > 2797 [i]