Best Known (112−85, 112, s)-Nets in Base 5
(112−85, 112, 51)-Net over F5 — Constructive and digital
Digital (27, 112, 51)-net over F5, using
- t-expansion [i] based on digital (22, 112, 51)-net over F5, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 22 and N(F) ≥ 51, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
(112−85, 112, 55)-Net over F5 — Digital
Digital (27, 112, 55)-net over F5, using
- t-expansion [i] based on digital (23, 112, 55)-net over F5, using
- net from sequence [i] based on digital (23, 54)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 23 and N(F) ≥ 55, using
- net from sequence [i] based on digital (23, 54)-sequence over F5, using
(112−85, 112, 251)-Net over F5 — Upper bound on s (digital)
There is no digital (27, 112, 252)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(5112, 252, F5, 85) (dual of [252, 140, 86]-code), but
- construction Y1 [i] would yield
- OA(5111, 143, S5, 85), but
- the linear programming bound shows that M ≥ 220 445839 236062 243251 966630 886846 194373 438661 017264 989094 781670 909972 692840 028685 168363 153934 478759 765625 / 378 761134 914432 479453 277033 > 5111 [i]
- linear OA(5140, 252, F5, 109) (dual of [252, 112, 110]-code), but
- discarding factors / shortening the dual code would yield linear OA(5140, 239, F5, 109) (dual of [239, 99, 110]-code), but
- construction Y1 [i] would yield
- linear OA(5139, 164, F5, 109) (dual of [164, 25, 110]-code), but
- construction Y1 [i] would yield
- OA(5138, 148, S5, 109), but
- the linear programming bound shows that M ≥ 86 158555 819318 512642 331328 682663 835348 540649 443889 196923 328992 021307 026760 723601 910285 651683 807373 046875 / 29 425011 > 5138 [i]
- OA(525, 164, S5, 16), but
- discarding factors would yield OA(525, 148, S5, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 313081 833470 800945 > 525 [i]
- discarding factors would yield OA(525, 148, S5, 16), but
- OA(5138, 148, S5, 109), but
- construction Y1 [i] would yield
- OA(599, 239, S5, 75), but
- discarding factors would yield OA(599, 147, S5, 75), but
- the linear programming bound shows that M ≥ 10 465163 072971 841494 226679 044239 098519 996159 375872 949750 717770 995481 826038 130942 490580 153891 865933 246663 189493 119716 644287 109375 / 6623 020670 394833 160024 516013 529216 797975 034523 571498 254336 > 599 [i]
- discarding factors would yield OA(599, 147, S5, 75), but
- linear OA(5139, 164, F5, 109) (dual of [164, 25, 110]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(5140, 239, F5, 109) (dual of [239, 99, 110]-code), but
- OA(5111, 143, S5, 85), but
- construction Y1 [i] would yield
(112−85, 112, 260)-Net in Base 5 — Upper bound on s
There is no (27, 112, 261)-net in base 5, because
- 1 times m-reduction [i] would yield (27, 111, 261)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 411954 426073 740232 964072 329639 302240 836194 144317 280694 447655 717872 606672 021625 > 5111 [i]