Best Known (165−145, 165, s)-Nets in Base 8
(165−145, 165, 65)-Net over F8 — Constructive and digital
Digital (20, 165, 65)-net over F8, using
- t-expansion [i] based on digital (14, 165, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(165−145, 165, 76)-Net over F8 — Digital
Digital (20, 165, 76)-net over F8, using
- net from sequence [i] based on digital (20, 75)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 20 and N(F) ≥ 76, using
(165−145, 165, 169)-Net over F8 — Upper bound on s (digital)
There is no digital (20, 165, 170)-net over F8, because
- 1 times m-reduction [i] would yield digital (20, 164, 170)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(8164, 170, F8, 144) (dual of [170, 6, 145]-code), but
- construction Y1 [i] would yield
- linear OA(8163, 166, F8, 144) (dual of [166, 3, 145]-code), but
- OA(86, 170, S8, 4), but
- discarding factors would yield OA(86, 104, S8, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 263173 > 86 [i]
- discarding factors would yield OA(86, 104, S8, 4), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(8164, 170, F8, 144) (dual of [170, 6, 145]-code), but
(165−145, 165, 296)-Net in Base 8 — Upper bound on s
There is no (20, 165, 297)-net in base 8, because
- 40 times m-reduction [i] would yield (20, 125, 297)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8125, 297, S8, 105), but
- 3 times code embedding in larger space [i] would yield OA(8128, 300, S8, 105), but
- the linear programming bound shows that M ≥ 6793 585140 889008 030729 322508 964555 065459 277460 834953 588076 472200 756209 612412 283054 591106 981448 232806 430680 691657 332122 308929 340323 182486 925159 649598 200067 760165 508386 184935 528421 176355 813423 447900 050657 116357 244986 094903 686745 957578 607461 012854 150912 811636 026330 225718 054844 338243 121606 759723 663854 185499 598087 080276 651536 482746 211810 578857 967991 785298 345770 463188 360626 060045 735281 983801 295511 929047 084219 565296 359370 532380 729461 956924 887302 682444 342143 504844 979062 286078 809895 271971 619883 764218 454071 889452 939836 361812 102422 008738 578116 448441 771297 490905 327557 219021 263329 572788 633600 / 130 611525 343440 163231 374769 973245 774840 656834 521906 614556 645656 647138 639117 056036 304540 496248 966375 052312 187931 915888 212123 495998 174664 040584 180534 466157 601711 026610 509898 479010 026641 238334 080982 196991 598646 853200 247200 324081 939757 560606 896186 803404 306589 734902 231907 936704 460297 438564 694639 031073 081365 071986 708664 761156 012294 110068 538636 589007 980478 580279 991605 083907 689585 005952 741207 946207 506656 972447 085962 852297 319551 612212 852231 239207 781658 035963 227143 534332 786569 > 8128 [i]
- 3 times code embedding in larger space [i] would yield OA(8128, 300, S8, 105), but
- extracting embedded orthogonal array [i] would yield OA(8125, 297, S8, 105), but