Arcs in the projective Hjelmslev Planes

This is a table of best known sizes n of u-arcs (at most u points on a line) in projective Hjelmslev planes of small sizes. A click on the best size of an u-arc in the tables gives more information on the found arcs.

Projective Hjelmslev plane PHG(2,Z8)

new: January 17, 2010

u	2	3	4	5	6	7	8	9	10	11	12
n	10	21	29	38	48	56	68	79	88	100	112

Projective Hjelmslev plane PHG(2,Z9)

new: September 23, 2009
All these arcs are optimal, i.e. of maximal possible size.

u	2	3	4	5	6	7	8	9	10	11	12
n	9	19	30	39	49	60	69	81	93	105	117

Projective Hjelmslev plane PHG(2,F3[X]/(X²))

new: September 23, 2009
All these arcs are optimal, i.e. of maximal possible size.

u	2	3	4	5	6	7	8	9	10	11	12
n	9	18	30	38	50	60	69	81	93	105	117

Projective Hjelmslev plane PHG(2,GR(16,4))

new: Spetember 27, 2010

u	2	3	4	5	6	7	8	9	10	11	12
n	optimal: 21	29	52	68	84	97	optimal: 126	140	152	166	186
u	13	14	15	16	17	18	19	20	21	22	23
n	203	224	236	256	276	296	316	336

Projective Hjelmslev plane PHG(2,F4[X]/(X²))

new: Oct 12, 2009

u	2	3	4	5	6	7	8	9	10	11	12
n	18	29	52	68	81	99	120	140	152	166	186
u	13	14	15	16	17	18	19	20
n	202	216	236	256	276	296	316	336

Projective Hjelmslev plane PHG(2,F4[X;x->x^2]/(X²))

new: Oct 7, 2009

u	2	3	4	5	6	7	8	9	10	11	12
n	18	29	52	68	81	96	120	140	152	166	186
u	13	14	15	16	17	18	19	20
n	202	219	236	256	276	296	316	336

Projective Hjelmslev plane PHG(2,Z16)

new: November 10, 2010

u	2	3	4	5	6	7	8	9	10	11	12
n	16	33	51	70	86	105	126	144	164	180	208
u	13	14	15	16	17	18	19	20	21	22	23
n	216	235	256	280	300	320	336	360	377	400	424

Projective Hjelmslev plane PHG(2,Z25)

new: October 29, 2010

u	2	3	4	5	6	7	8	9	10	11	12
n	21 optimal	40	66	87	114	142	162	186	210	235	264
u	13	14	15	16	17	18	19	20	21	22	23
n	310	319	350	372	400	425	465	481	510	540	566
u	24	25	26	27	28	29	30
n	595	625	655	685	715	745	775

Projective Hjelmslev plane PHG(2,F5[X]/(X²))

new: Oct 8, 2009

u	2	3	4	5	6	7	8	9	10	11	12
n	25	42	64	90	130	152	162	190	225	250	280
u	13	14	15	16	17	18	19	20	21	22	23
n	297	318	346	375	405	433	454	490	511	540	565
u	24	25	26	27	28	29	30
n	595	625	655	685	715	745	775

Projective Hjelmslev plane PHG(2,Z27)

new: January 31, 2011

u	2	3	4	5	6	7	8	9	10	11	12
n	21	46	72	95	126	149	177	208	234	260	289
u	13	14	15	16	17	18	19	20	21	22	23
n	318	351	374	412	441	469	505	534	567	603	636
u	24	25	26	27	28	29	30
n

Projective Hjelmslev plane PHG(2,Z32)

new: May 21, 2010

u	2	3	4	5	6	7	8	9	10	11	12
n	26	53	84	115	147	182	224	250	283	319	352
u	13	14	15	16	17	18	19	20	21	22	23
n	388	427