On the Minimal Weight of vectors in k-dimensional subspaces of GF(q)^n for k\ from 1 to, 5, q=, 2, and n from , 10, to , 100, in increments of, 10 By Shalosh B. Ekhad This article reports simulating the Calabi-Wilf algorithm, 1000, times for in\ vestigating the minimum weight of vectors in low-dimensional subspaces o\ f GF(q)^n with q=, 2 We run each simulation twice, and for each report the smallest smallest weig\ ht recorded in the sample (offering upper bound for the smallest possibl\ e smallest weight followed by the largest smallest weight, offering lower bounds for this impo\ rtant quantity followed by the estimated average, variance, and scaled central moments up t\ o the, 4 Investigating n=, 10 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 10, the numbers are: [1, 9, [4.958000000, 2.348236000, -.8282736699e-1, 2.714997214]] and again: [1, 9, [5.034000000, 2.364844000, .4035193133e-1, 2.800478242]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 10, the numbers are: [1, 6, [3.632000000, 1.222576000, -.1778660399, 2.885099776]] and again: [1, 6, [3.582000000, 1.245276000, -.5712988484e-1, 2.661522069]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 10, the numbers are: [1, 5, [2.860000000, .7584000000, -.1615961035, 2.469980939]] and again: [1, 5, [2.882000000, .7860760000, -.6038672905e-1, 2.515343316]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 10, the numbers are: [1, 4, [2.312000000, .5926560000, -.1993933127e-1, 2.487546337]] and again: [1, 4, [2.349000000, .5991990000, -.4499495170e-1, 2.499448727]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 10, the numbers are: [1, 3, [1.893000000, .3935510000, .8325573013e-1, 2.496293040]] and again: [1, 4, [1.860000000, .3844000000, .1266825551, 2.621204066]] Investigating n=, 20 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 20, the numbers are: [2, 17, [10.03100000, 4.988039000, .1033265007, 2.864629291]] and again: [4, 16, [9.951000000, 4.960599000, .3659570915e-1, 2.640279087]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 20, the numbers are: [2, 12, [8.077000000, 2.559071000, -.3338247466, 3.120984506]] and again: [3, 12, [8.063000000, 2.665031000, -.2640853406, 2.941974691]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 20, the numbers are: [2, 10, [6.979000000, 1.768559000, -.4079885936, 3.127985569]] and again: [2, 10, [6.897000000, 1.820391000, -.2986054348, 3.087928235]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 20, the numbers are: [2, 10, [6.181000000, 1.396239000, -.4116101740, 3.219895661]] and again: [2, 9, [6.073000000, 1.427671000, -.4854156930, 3.248439330]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 20, the numbers are: [2, 8, [5.435000000, 1.083775000, -.2762125819, 2.766107671]] and again: [2, 8, [5.389000000, 1.105679000, -.4026982223, 3.179263823]] Investigating n=, 30 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 30, the numbers are: [6, 23, [14.95700000, 7.297151000, -.4646314919e-2, 2.866832663]] and again: [8, 23, [14.93400000, 7.613644000, .5492088766e-1, 2.626909807]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 30, the numbers are: [5, 18, [12.65800000, 3.997036000, -.3580277961, 3.140988323]] and again: [6, 18, [12.57000000, 4.255100000, -.2936429152, 2.932310590]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 30, the numbers are: [5, 16, [11.30300000, 2.983191000, -.3599269126, 3.129523344]] and again: [3, 16, [11.21800000, 2.818476000, -.5244666627, 3.645111626]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 30, the numbers are: [5, 14, [10.19900000, 2.217399000, -.4229873215, 2.964558069]] and again: [5, 14, [10.27700000, 2.042271000, -.4157451414, 3.118133118]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 30, the numbers are: [5, 13, [9.481000000, 1.629639000, -.4854241228, 3.163063442]] and again: [4, 13, [9.491000000, 1.735919000, -.5563171701, 3.484037699]] Investigating n=, 40 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 40, the numbers are: [10, 30, [20.07400000, 9.734524000, -.6655721236e-1, 2.972853957]] and again: [11, 30, [20.11700000, 10.45331100, -.4248024960e-1, 2.933203405]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 40, the numbers are: [9, 25, [17.28900000, 5.415479000, -.2477578154, 3.226866047]] and again: [10, 24, [17.20900000, 5.211319000, -.1761539399, 2.963976697]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 40, the numbers are: [7, 21, [15.67500000, 3.991375000, -.2918285301, 3.265762406]] and again: [9, 21, [15.68000000, 3.911600000, -.4746597109, 3.135295261]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 40, the numbers are: [8, 19, [14.51900000, 2.905639000, -.5497506403, 3.347627868]] and again: [8, 18, [14.57300000, 2.878671000, -.5338848388, 3.431604677]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 40, the numbers are: [8, 17, [13.52900000, 2.199159000, -.5081174293, 3.316300697]] and again: [8, 17, [13.53200000, 2.278976000, -.5469746197, 3.278638647]] Investigating n=, 50 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 50, the numbers are: [14, 36, [24.97800000, 11.71151600, .5616298327e-1, 2.935937800]] and again: [16, 35, [24.91500000, 11.92977500, .1863146393e-1, 2.844056666]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 50, the numbers are: [15, 30, [21.89600000, 6.077184000, -.1558457414, 2.886433081]] and again: [13, 29, [21.88700000, 6.876231000, -.2945173851, 2.938201526]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 50, the numbers are: [12, 26, [20.10000000, 4.872000000, -.3314743162, 3.250849835]] and again: [13, 26, [20.34800000, 4.570896000, -.3784283305, 3.112983695]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 50, the numbers are: [12, 23, [18.85900000, 3.347119000, -.5154428862, 3.322544594]] and again: [12, 23, [18.81200000, 3.352656000, -.4535833216, 3.215025755]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 50, the numbers are: [10, 23, [17.74800000, 3.010496000, -.6386707601, 3.874193057]] and again: [11, 22, [17.80300000, 2.724191000, -.5500860184, 3.705881482]] Investigating n=, 60 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 60, the numbers are: [18, 41, [30.17700000, 14.31167100, -.3990283445e-2, 2.707402222]] and again: [19, 43, [30.05600000, 14.96886400, .5213000836e-1, 3.073484303]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 60, the numbers are: [16, 35, [26.74000000, 8.534400000, -.3417112729, 3.032560168]] and again: [18, 35, [26.75400000, 8.187484000, -.2024438205, 2.876958805]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 60, the numbers are: [16, 30, [24.61200000, 5.351456000, -.3100651413, 2.972692726]] and again: [17, 31, [24.78600000, 5.784204000, -.4235960757, 3.001294453]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 60, the numbers are: [15, 29, [23.20500000, 4.402975000, -.5218646326, 3.278344647]] and again: [15, 28, [23.28300000, 3.912911000, -.4095169227, 3.517920324]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 60, the numbers are: [14, 27, [22.00700000, 3.612951000, -.4084322137, 3.160128935]] and again: [14, 27, [22.01900000, 3.802639000, -.6108678726, 3.743484627]] Investigating n=, 70 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 70, the numbers are: [22, 48, [34.83500000, 17.76177500, .5498129386e-2, 2.992128063]] and again: [22, 48, [35.25600000, 17.60646400, -.2523851833e-1, 2.880405298]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 70, the numbers are: [22, 40, [31.45700000, 9.686151000, -.2159556697, 2.865934041]] and again: [18, 41, [31.39600000, 9.961184000, -.2709383394, 3.070283566]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 70, the numbers are: [21, 37, [29.34700000, 5.984591000, -.2866556267, 2.918526461]] and again: [20, 36, [29.37400000, 6.670124000, -.2356925146, 2.947649089]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 70, the numbers are: [20, 34, [27.69300000, 4.868751000, -.4199886327, 3.444720796]] and again: [19, 33, [27.70700000, 5.329151000, -.3881806276, 3.157823110]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 70, the numbers are: [19, 31, [26.27600000, 4.105824000, -.3073054406, 2.830865838]] and again: [19, 31, [26.45800000, 3.726236000, -.4479290617, 3.192027610]] Investigating n=, 80 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 80, the numbers are: [26, 55, [40.03500000, 18.89777500, .5339751028e-1, 3.262854818]] and again: [26, 53, [39.89000000, 18.71590000, .7355675689e-2, 2.958754704]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 80, the numbers are: [21, 45, [36.12500000, 11.55737500, -.4613053157, 3.592492756]] and again: [26, 45, [36.21100000, 10.44047900, -.1518665499, 2.953169295]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 80, the numbers are: [21, 40, [33.97000000, 7.311100000, -.5418703946, 3.994057297]] and again: [25, 41, [33.87500000, 7.559375000, -.4422771156, 3.218210833]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 80, the numbers are: [22, 38, [32.30600000, 6.160364000, -.5804986488, 3.519473106]] and again: [24, 39, [32.24700000, 5.735991000, -.4650311341, 3.341886017]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 80, the numbers are: [23, 36, [30.93600000, 4.523904000, -.5648606571, 3.176895112]] and again: [23, 36, [30.89100000, 4.203119000, -.4364944436, 3.146080406]] Investigating n=, 90 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 90, the numbers are: [28, 57, [45.26000000, 24.45240000, -.3812948073e-1, 2.841411534]] and again: [28, 60, [44.97700000, 20.82647100, .1013011316, 3.281791994]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 90, the numbers are: [28, 51, [40.98000000, 11.89160000, -.2828890389, 3.100791694]] and again: [28, 52, [40.91700000, 11.91811100, -.1338897500, 2.978789927]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 90, the numbers are: [27, 46, [38.69500000, 8.463975000, -.4698672642, 3.472649120]] and again: [29, 45, [38.67400000, 8.593724000, -.3552912471, 2.960960982]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 90, the numbers are: [25, 43, [36.63500000, 6.375775000, -.4197086980, 3.349077613]] and again: [26, 43, [36.84700000, 5.903591000, -.6605953947, 3.928401234]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 90, the numbers are: [26, 40, [35.19600000, 5.333584000, -.6221691591, 3.674097183]] and again: [27, 41, [35.19200000, 4.893136000, -.3957494290, 3.033069030]] Investigating n=, 100 For , 1, dim. subspaces of GF(q)^n with q=, 2, with n= , 100, the numbers are: [33, 65, [49.86600000, 25.67404400, .6928605972e-1, 2.971926578]] and again: [33, 66, [49.65100000, 24.61519900, .3410945823e-1, 2.934102823]] For , 2, dim. subspaces of GF(q)^n with q=, 2, with n= , 100, the numbers are: [36, 57, [45.96000000, 12.48240000, -.2433976551, 2.848173009]] and again: [32, 56, [45.58100000, 13.41143900, -.1662482222, 2.873528331]] For , 3, dim. subspaces of GF(q)^n with q=, 2, with n= , 100, the numbers are: [33, 51, [43.28200000, 9.630476000, -.4377866843, 3.036884411]] and again: [31, 51, [43.32100000, 9.227959000, -.4501452925, 3.359717614]] For , 4, dim. subspaces of GF(q)^n with q=, 2, with n= , 100, the numbers are: [31, 49, [41.24900000, 7.322999000, -.4483428107, 3.418660002]] and again: [30, 48, [41.25300000, 6.954991000, -.4492458390, 3.522679991]] For , 5, dim. subspaces of GF(q)^n with q=, 2, with n= , 100, the numbers are: [29, 45, [39.78700000, 5.833631000, -.6786162225, 3.855351202]] and again: [29, 45, [39.72900000, 6.185559000, -.5591689205, 3.483511941]] ------------------------------------------------------ This ends this article that took, 586.062, to generate