Best Known (223−51, 223, s)-Nets in Base 4
(223−51, 223, 1044)-Net over F4 — Constructive and digital
Digital (172, 223, 1044)-net over F4, using
- 1 times m-reduction [i] based on digital (172, 224, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 56, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 56, 261)-net over F256, using
(223−51, 223, 3171)-Net over F4 — Digital
Digital (172, 223, 3171)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4223, 3171, F4, 51) (dual of [3171, 2948, 52]-code), using
- 2947 step Varšamov–Edel lengthening with (ri) = (14, 6, 4, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 25 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 29 times 0, 1, 31 times 0, 1, 31 times 0, 1, 33 times 0, 1, 33 times 0, 1, 35 times 0, 1, 35 times 0, 1, 37 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 42 times 0, 1, 43 times 0, 1, 45 times 0, 1, 46 times 0, 1, 48 times 0, 1, 49 times 0, 1, 50 times 0, 1, 51 times 0, 1, 53 times 0, 1, 55 times 0, 1, 57 times 0, 1, 58 times 0, 1, 59 times 0, 1, 62 times 0, 1, 63 times 0, 1, 65 times 0, 1, 67 times 0, 1, 69 times 0, 1, 71 times 0, 1, 73 times 0, 1, 75 times 0, 1, 77 times 0, 1, 79 times 0, 1, 82 times 0, 1, 84 times 0) [i] based on linear OA(451, 52, F4, 51) (dual of [52, 1, 52]-code or 52-arc in PG(50,4)), using
- dual of repetition code with length 52 [i]
- 2947 step Varšamov–Edel lengthening with (ri) = (14, 6, 4, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 25 times 0, 1, 27 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 29 times 0, 1, 31 times 0, 1, 31 times 0, 1, 33 times 0, 1, 33 times 0, 1, 35 times 0, 1, 35 times 0, 1, 37 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 42 times 0, 1, 43 times 0, 1, 45 times 0, 1, 46 times 0, 1, 48 times 0, 1, 49 times 0, 1, 50 times 0, 1, 51 times 0, 1, 53 times 0, 1, 55 times 0, 1, 57 times 0, 1, 58 times 0, 1, 59 times 0, 1, 62 times 0, 1, 63 times 0, 1, 65 times 0, 1, 67 times 0, 1, 69 times 0, 1, 71 times 0, 1, 73 times 0, 1, 75 times 0, 1, 77 times 0, 1, 79 times 0, 1, 82 times 0, 1, 84 times 0) [i] based on linear OA(451, 52, F4, 51) (dual of [52, 1, 52]-code or 52-arc in PG(50,4)), using
(223−51, 223, 752983)-Net in Base 4 — Upper bound on s
There is no (172, 223, 752984)-net in base 4, because
- 1 times m-reduction [i] would yield (172, 222, 752984)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 45 427890 725765 716700 234084 098369 091693 738409 066148 003735 055149 495852 363678 667139 019607 675955 509738 072437 298164 837166 779238 665525 024216 > 4222 [i]