Best Known (241−178, 241, s)-Nets in Base 4
(241−178, 241, 66)-Net over F4 — Constructive and digital
Digital (63, 241, 66)-net over F4, using
- t-expansion [i] based on digital (49, 241, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(241−178, 241, 99)-Net over F4 — Digital
Digital (63, 241, 99)-net over F4, using
- t-expansion [i] based on digital (61, 241, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(241−178, 241, 301)-Net over F4 — Upper bound on s (digital)
There is no digital (63, 241, 302)-net over F4, because
- 2 times m-reduction [i] would yield digital (63, 239, 302)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4239, 302, F4, 176) (dual of [302, 63, 177]-code), but
- residual code [i] would yield OA(463, 125, S4, 44), but
- the linear programming bound shows that M ≥ 175234 933305 074206 543921 418495 881277 780883 332813 429919 101006 437404 404381 504833 864636 079141 500337 375376 052360 191696 611094 696757 939320 665090 885499 385380 009810 778380 200900 617655 056099 806688 415743 521650 982924 518318 871643 642600 732269 143252 854005 385711 951077 298974 645803 648886 013429 809152 / 2025 517705 388844 186066 624345 865386 010728 459225 322617 301863 147239 853832 944815 211835 885283 966428 532227 973928 412742 757997 253523 507788 055914 588259 352119 823340 513122 117611 644548 283151 687312 224909 613727 897424 811994 051452 501775 130990 943683 258327 507029 > 463 [i]
- residual code [i] would yield OA(463, 125, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4239, 302, F4, 176) (dual of [302, 63, 177]-code), but
(241−178, 241, 413)-Net in Base 4 — Upper bound on s
There is no (63, 241, 414)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 14 907257 191607 746071 971412 139800 257896 827041 792718 896559 114159 450307 730368 049710 300094 258576 025523 141468 719775 819993 631310 444015 027404 778665 114506 > 4241 [i]