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The Hunt for Weighing Matrices of Small Orders

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Algebraic Design Theory and Hadamard Matrices

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 133))

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Abstract

In this note we use a variety of techniques to construct new weighing matrices of small orders. In particular, we construct new examples of W(n, 9) for n ∈ { 14, 18, 19, 21} and W(n, n − 1) for n ∈ { 42, 46}. We also discuss two possible approaches for constructing a W(66, 65), and show nonexistence of these under certain assumptions.

This paper is in final form and no similar paper has been or is being submitted elsewhere.

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Acknowledgements

We are greatly indebted to Prof. A. Munemasa who pointed out that certain maximal self-orthogonal ternary codes of length 19 (see [10]) contain symmetric W(19, 9) matrices, and provided us with such an example.

This work was supported in part by the JSPS KAKENHI Grant Numbers 24 ⋅ 02807. Part of the computational results in this research were obtained using supercomputing resources at Cyberscience Center, Tohoku University.

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Authors and Affiliations

  1. Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Ferenc Szöllősi

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  1. Ferenc Szöllősi
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Correspondence to Ferenc Szöllősi .

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Editors and Affiliations

  1. School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, Arizona, USA

    Charles J. Colbourn

Additional information

Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

Appendices

Appendix 1: New Weighing Matrices of Orders n ∈ { 14, 18, 19, 21}

W14:={{0,0,0,1,0,-1,-1,1,1,0,-1,1,1,1},                                {0,0,-1,1,0,-1,0,-1,-1,0,1,-1,1,1},

      {0,-1,0,1,0,-1,0,1,-1,0,-1,-1,-1,-1},                              {1,1,1,0,-1,-1,1,0,1,1,0,-1,0,0},

      {0,0,0,-1,0,-1,0,-1,-1,1,-1,1,1,-1},                           {-1,-1,-1,-1,-1,0,-1,0,1,1,0,-1,0,0},

      {-1,0,0,1,0,-1,0,-1,1,0,1,1,-1,-1},                              {1,-1,1,0,-1,0,-1,0,0,-1,1,0,1,-1},

      {1,-1,-1,1,-1,1,1,0,0,1,0,1,0,0},                                 {0,0,0,1,1,1,0,-1,1,0,-1,-1,1,-1},

      {-1,1,-1,0,-1,0,1,1,0,-1,0,0,1,-1},                              {1,-1,-1,-1,1,-1,1,0,1,-1,0,0,0,0},

      {1,1,-1,0,1,0,-1,1,0,1,1,0,0,-1},                              {1,1,-1,0,-1,0,-1,-1,0,-1,-1,0,-1,0}};

W18:={{0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1},                   {0,0,-1,0,0,0,0,1,-1,0,1,-1,1,-1,0,-1,1,0},

      {0,1,0,0,-1,-1,1,-1,0,0,1,0,0,1,-1,-1,0,0},              {0,0,0,0,0,-1,-1,0,1,-1,0,1,1,0,1,-1,0,-1},

      {0,0,1,0,0,-1,-1,-1,-1,0,0,0,1,-1,-1,1,0,0},                {0,0,1,1,1,0,1,0,1,-1,1,0,0,-1,0,0,0,1},

      {0,0,-1,1,1,-1,0,-1,0,1,0,-1,0,0,1,0,-1,0},               {0,-1,1,0,1,0,1,0,-1,1,0,1,0,0,0,-1,0,-1},

      {0,1,0,-1,1,-1,0,1,0,0,1,0,-1,0,0,1,0,-1},               {-1,0,0,1,0,1,-1,-1,0,0,1,0,-1,0,0,0,1,-1},

      {-1,-1,-1,0,0,-1,0,0,-1,-1,0,1,-1,0,0,0,0,1},            {-1,1,0,-1,0,0,1,-1,0,0,-1,0,0,-1,1,0,1,0},

      {-1,-1,0,-1,-1,0,0,0,1,1,1,0,0,-1,0,0,-1,0},              {-1,1,-1,0,1,1,0,0,0,0,0,1,1,0,-1,0,-1,0},

      {-1,0,1,-1,1,0,-1,0,0,0,0,-1,0,1,0,-1,0,1},               {-1,1,1,1,-1,0,0,1,-1,0,0,0,0,0,1,0,-1,0},

      {-1,-1,0,0,0,0,1,0,0,-1,0,-1,1,1,0,1,0,-1},               {-1,0,0,1,0,-1,0,1,1,1,-1,0,0,0,-1,0,1,0}};

W19:={{0,-1,0,1,1,1,1,0,-1,0,0,0,-1,0,0,1,0,1,0},           {-1,1,0,0,0,0,0,-1,-1,-1,0,0,0,1,-1,0,-1,0,1},

      {0,0,0,-1,1,-1,-1,0,0,0,0,1,0,0,1,1,0,1,1},            {1,0,-1,0,0,0,0,1,-1,-1,0,1,1,-1,-1,0,0,0,0},

      {1,0,1,0,1,1,-1,1,0,0,0,0,0,1,0,0,-1,-1,0},           {1,0,-1,0,1,0,0,-1,1,-1,-1,0,-1,0,0,-1,0,0,0},

      {1,0,-1,0,-1,0,0,0,0,-1,1,-1,0,1,1,1,0,0,0},            {0,-1,0,1,1,-1,0,0,0,0,1,-1,1,0,0,-1,0,0,1},

      {-1,-1,0,-1,0,1,0,0,1,-1,1,0,0,-1,0,0,-1,0,0       {0,-1,0,-1,0,-1,-1,0,-1,0,0,-1,-1,0,-1,0,0,0,-1},

      {0,0,0,0,0,-1,1,1,1,0,0,0,-1,0,-1,1,0,-1,1},           {0,0,1,1,0,0,-1,-1,0,-1,0,0,0,-1,0,1,1,-1,0},

      {-1,0,0,1,0,-1,0,1,0,-1,-1,0,0,0,1,0,-1,0,-1},         {0,1,0,-1,1,0,1,0,-1,0,0,-1,0,-1,1,0,0,-1,0},

      {0,-1,1,-1,0,0,1,0,0,-1,-1,0,1,1,0,0,1,0,0},            {1,0,1,0,0,-1,1,-1,0,0,1,1,0,0,0,0,-1,0,-1},

      {0,-1,0,0,-1,0,0,0,-1,0,0,1,-1,0,1,-1,0,-1,1},         {1,0,1,0,-1,0,0,0,0,0,-1,-1,0,-1,0,0,-1,1,1},

      {0,1,1,0,0,0,0,1,0,-1,1,0,-1,0,0,-1,1,1,0}};

W21:={{0,0,1,1,-1,0,0,0,0,1,1,1,0,-1,0,0,1,0,1,0,0},   {0,0,0,-1,-1,0,-1,-1,0,1,-1,0,0,0,0,0,0,1,0,-1,-1},

      {1,0,0,0,1,0,1,-1,1,0,-1,0,-1,0,0,0,1,0,1,0,0},   {1,-1,0,0,-1,1,0,-1,0,-1,0,0,0,-1,0,0,0,0,-1,0,1},

      {-1,-1,1,-1,0,0,1,0,0,0,0,0,0,0,1,1,0,1,0,1,0},      {0,0,0,1,0,1,1,-1,-1,1,0,-1,1,1,0,0,0,0,0,0,0},

      {0,-1,1,0,1,1,0,0,0,0,0,1,0,0,0,0,-1,-1,0,-1,-1}, {0,-1,-1,-1,0,-1,0,-1,0,0,1,0,1,0,0,0,0,-1,1,0,0},

      {0,0,1,0,0,-1,0,0,-1,0,-1,-1,0,-1,-1,1,0,-1,0,0,0},  {1,1,0,-1,0,1,0,0,0,0,1,0,0,0,-1,1,0,0,0,1,-1},

      {1,-1,-1,0,0,0,0,1,-1,1,-1,1,0,0,0,0,0,0,0,1,0},   {1,0,0,0,0,-1,1,0,-1,0,1,0,-1,0,0,0,-1,1,0,-1,0},

      {0,0,-1,0,0,1,0,1,0,0,0,-1,0,-1,1,1,0,0,1,-1,0},  {-1,0,0,-1,0,1,0,0,-1,0,0,0,-1,0,-1,-1,0,0,1,0,1},

      {0,0,0,0,1,0,0,0,-1,-1,0,0,1,-1,0,-1,1,1,0,0,-1},    {0,0,0,0,1,0,0,0,1,1,0,0,1,-1,-1,0,-1,1,0,0,1},

      {1,0,1,0,0,0,-1,0,0,0,0,-1,0,0,1,-1,-1,0,1,1,0},     {0,1,0,0,1,0,-1,-1,-1,0,0,1,0,0,1,1,0,0,0,0,1},

      {1,0,1,-1,0,0,0,1,0,0,0,0,1,1,0,0,1,0,0,-1,1},     {0,-1,0,0,1,0,-1,0,0,1,1,-1,-1,0,0,0,1,0,-1,0,0},

      {0,-1,0,1,0,0,-1,0,0,-1,0,0,0,1,-1,1,0,1,1,0,0}};

Appendix 2: A New Conference Matrix of Order 42

We only present the matrix B leading to a W(42, 41) via formulae (2) and (4).

B:={{1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1},  {-1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, -1, 1},

    {-1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, -1, 1, -1}, {-1, 1, -1, -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, -1},

    {-1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1},  {1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, -1},

    {-1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1},    {-1, -1, 1, -1, 1, 1, 1, 1, 1, -1, -1, 1, -1, -1},

    {1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, 1},   {1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, -1, 1, 1},

    {1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1},     {1, 1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, -1},

    {-1, 1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1},

    {-1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, -1, 1}};

Appendix 3: A New Conference Matrix of Order 46

Here we present the matrices A(x) and B(y) leading to 12 distinct W(46, 45) matrices via formula (8).

Ax:={{0,1,-1,1,1,-1,1,-1,1,-1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,1,1},

     {1,0,-1,-1,-1,1,1,-1,1,-1,-1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1},

     {-1,-1,0,1,-1,1,1,-1,-1,1,-1,-1,-1,-1,1,1,-1,-1,-1,1,1,1,1},

     {1,-1,1,0,1,1,-1,-1,-1,1,-1,-1,-1,-1,-1,-1,1,1,1,-1,1,1,1},

     {1,-1,-1,1,0,1,-1,1,-1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,-1,1,-1},

     {-1,1,1,1,1,0,-1,-1,1,1,1,-1,-1,1,-1,1,-1,1,1,1,1,-1,-1},

     {1,1,1,-1,-1,-1,0,1,-1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,-1,-1},

     {-1,-1,-1,-1,1,-1,1,0,1,-1,-1,1,-1,1,1,1,1,-1,-1,-1,1,-1,-1},

     {1,1,-1,-1,-1,1,-1,1,0,1,-1,-1,1,1,1,-1,-1,1,-1,-1,1,-1,-1},

     {-1,-1,1,1,-1,1,-1,-1,1,0,1,1,-1,1,1,x[1],x[14],x[27],x[40],x[53],x[66],x[79],x[92]},

     {-1,-1,-1,-1,1,1,1,-1,-1,1,0,1,1,1,1,x[2],x[15],x[28],x[41],x[54],x[67],x[80],x[93]},

     {1,1,-1,-1,1,-1,1,1,-1,1,1,0,1,-1,-1,x[3],x[16],x[29],x[42],x[55],x[68],x[81],x[94]},

     {-1,-1,-1,-1,-1,-1,1,-1,1,-1,1,1,0,1,-1,x[4],x[17],x[30],x[43],x[56],x[69],x[82],x[95]},

     {-1,-1,-1,-1,1,1,-1,1,1,1,1,-1,1,0,1,x[5],x[18],x[31],x[44],x[57],x[70],x[83],x[96]},

     {1,-1,1,-1,-1,-1,1,1,1,1,1,-1,-1,1,0,x[6],x[19],x[32],x[45],x[58],x[71],x[84],x[97]},

     {1,-1,1,-1,1,1,-1,1,-1,x[1],x[2],x[3],x[4],x[5],x[6],0,x[20],x[33],x[46],x[59],x[72],x[85],x[98]},

     {-1,1,-1,1,-1,-1,1,1,-1,x[14],x[15],x[16],x[17],x[18],x[19],x[20],0,x[34],x[47],x[60],x[73],x[86],

                                                                                                   x[99]},

     {-1,1,-1,1,1,1,-1,-1,1,x[27],x[28],x[29],x[30],x[31],x[32],x[33],x[34],0,x[48],x[61],x[74],x[87],

                                                                                                  x[100]},

     {1,1,-1,1,-1,1,1,-1,-1,x[40],x[41],x[42],x[43],x[44],x[45],x[46],x[47],x[48],0,x[62],x[75],x[88],

                                                                                                  x[101]},

     {-1,1,1,-1,-1,1,1,-1,-1,x[53],x[54],x[55],x[56],x[57],x[58],x[59],x[60],x[61],x[62],0,x[76],x[89],

                                                                                                  x[102]},

     {1,-1,1,1,-1,1,1,1,1,x[66],x[67],x[68],x[69],x[70],x[71],x[72],x[73],x[74],x[75],x[76],0,x[90],

                                                                                                  x[103]},

     {1,-1,1,1,1,-1,-1,-1,-1,x[79],x[80],x[81],x[82],x[83],x[84],x[85],x[86],x[87],x[88],x[89],x[90],0,

                                                                                                  x[104]},

     {1,-1,1,1,-1,-1,-1,-1,-1,x[92],x[93],x[94],x[95],x[96],x[97],x[98],x[99],x[100],x[101],x[102],x[103],

                                                                                                x[104],0}};

By:={{1,1,-1,-1,-1,1,1,-1,1,1,-1,1,-1,-1,-1,1,-1,1,-1,-1,1,1,-1},

     {1,1,1,1,-1,-1,1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,1,1,1,1,1},

     {1,1,1,-1,1,-1,1,1,-1,1,1,-1,1,1,-1,-1,-1,1,-1,1,-1,-1,1},

     {1,-1,1,-1,-1,1,-1,1,1,-1,1,1,1,1,-1,-1,-1,-1,1,-1,1,1,-1},

     {1,1,1,1,1,-1,1,-1,1,-1,-1,1,1,1,-1,1,-1,-1,-1,1,1,-1,-1},

     {1,1,-1,1,-1,1,1,1,-1,-1,1,-1,1,-1,1,1,-1,-1,-1,-1,-1,-1,-1},

     {1,-1,1,-1,1,1,1,1,1,1,-1,-1,1,1,-1,1,-1,-1,-1,-1,-1,-1,-1},

     {1,-1,-1,1,1,1,1,1,-1,-1,1,1,1,1,-1,-1,1,-1,1,-1,1,-1,1},

     {1,1,-1,-1,-1,-1,1,1,1,1,-1,1,1,-1,1,-1,-1,1,1,-1,1,-1,1},

     {1,-1,1,-1,-1,-1,1,1,1,1,-1,1,-1,-1,1,x[105],x[133],x[161],x[189],x[217],x[245],x[273],x[301]},

     {1,-1,-1,1,1,1,-1,-1,1,1,1,-1,1,-1,-1,x[107],x[135],x[163],x[191],x[219],x[247],x[275],x[303]},

     {1,1,-1,-1,-1,-1,1,1,-1,-1,1,1,-1,1,-1,x[109],x[137],x[165],x[193],x[221],x[249],x[277],x[305]},

     {1,1,1,-1,-1,1,-1,1,1,-1,-1,-1,1,1,1,x[111],x[139],x[167],x[195],x[223],x[251],x[279],x[307]},

     {1,1,1,-1,-1,1,1,-1,-1,-1,-1,-1,-1,1,-1,x[113],x[141],x[169],x[197],x[225],x[253],x[281],x[309]},

     {1,-1,-1,1,-1,-1,-1,1,1,-1,-1,-1,-1,1,-1,x[115],x[143],x[171],x[199],x[227],x[255],x[283],x[311]},

     {1,-1,-1,1,-1,-1,1,-1,-1,x[106],x[108],x[110],x[112],x[114],x[116],x[118],x[145],x[173],x[201],x[229],

                                                                                     x[257],x[285],x[313]},

     {1,-1,-1,-1,1,-1,-1,1,1,x[134],x[136],x[138],x[140],x[142],x[144],x[146],x[148],x[175],x[203],x[231],

                                                                                    x[259],x[287],x[315]},

     {1,-1,-1,-1,1,-1,1,-1,-1,x[162],x[164],x[166],x[168],x[170],x[172],x[174],x[176],x[178],x[205],x[233],

                                                                                     x[261],x[289],x[317]},

     {1,-1,1,1,-1,1,-1,-1,-1,x[190],x[192],x[194],x[196],x[198],x[200],x[202],x[204],x[206],x[208],x[235],

                                                                                     x[263],x[291],x[319]},

     {1,1,1,1,1,-1,-1,-1,-1,x[218],x[220],x[222],x[224],x[226],x[228],x[230],x[232],x[234],x[236],x[238],

                                                                                     x[265],x[293],x[321]},

     {1,1,-1,-1,1,1,-1,-1,-1,x[246],x[248],x[250],x[252],x[254],x[256],x[258],x[260],x[262],x[264],x[266],

                                                                                     x[268],x[295],x[323]},

     {1,1,-1,-1,1,1,-1,1,-1,x[274],x[276],x[278],x[280],x[282],x[284],x[286],x[288],x[290],x[292],x[294],

                                                                                     x[296],x[298],x[325]},

     {1,1,-1,-1,1,1,1,-1,1,x[302],x[304],x[306],x[308],x[310],x[312],x[314],x[316],x[318],x[320],x[322],

                                                                                    x[324],x[326],x[328]}};

The following is the relevant Gröbner basis:

GB:={-1+x[1],-1+x[98]+x[99]+x[100],1+x[101],-1+x[102],1+x[103],1+x[104],1+x[105],1+x[106],-1+x[107],

     1+x[108],1+x[109],1+x[110],1+x[111],-1+x[112],-1+x[113],1+x[114],-1+x[115],-1+x[116],-1+x[118],

     -1+x[133],1+x[134],1+x[135],1+x[136],-1+x[137],1+x[138],-1+x[139],-1+x[14],-1+x[140],-1+x[141],

     1+x[142],1+x[143],-1+x[144],-1+x[145],-1+x[146],-1+x[148],1+x[15],-1+x[16],1+x[161],1+x[162],

     -1+x[163],1+x[164],-1+x[165],1+x[166],-1+x[167],-1+x[168],1+x[169],1+x[17],-1+x[170],-1+x[171],

     -1+x[172],-1+x[173],1+x[174],1+x[175],1+x[176],-1+x[178],1+x[18],-1+x[189],1+x[19],1+x[190],

     -1+x[191],1+x[192],1+x[193],-1+x[194],1+x[195],1+x[196],-1+x[197],x[94]+x[198],-1+x[199],1+x[2],

     -1+x[20],x[94]+x[200],1+x[201],x[94]+x[202],-1+x[203],-x[87]+x[204],-1+x[205],-x[86]+x[206],

     -1+x[86]+x[87]+x[208],-x[94]+x[217],1+x[218],-x[94]+x[219],1+x[220],-x[94]+x[221],-1+x[222],

     -x[86]-x[87]+x[98]+x[223],1+x[224],-1+x[86]+x[99]+x[225],-x[94]+x[226],x[87]-x[98]-x[99]+x[227],

     -x[94]+x[228],-x[86]-x[87]+x[98]+x[229],-x[94]+x[230],-1+x[86]+x[99]+x[231],-1+x[98]+x[99]+x[232],

     x[87]-x[98]-x[99]+x[233],-x[99]+x[234],1+x[235],-x[98]+x[236],1+x[238],1+x[245],1+x[246],1+x[247],

     1+x[248],1+x[249],-1+x[250],-x[98]+x[251],-1+x[252],-x[99]+x[253],1+x[254],-1+x[98]+x[99]+x[255],

     1+x[256],-x[98]+x[257],1+x[258],-x[99]+x[259],x[87]-x[98]-x[99]+x[260],-1+x[98]+x[99]+x[261],

     -1+x[86]+x[99]+x[262],1+x[263],-x[86]-x[87]+x[98]+x[264],-1+x[265],-1+x[266],1+x[268],1+x[27],

     x[94]+x[273],-1+x[274],x[94]+x[275],1+x[276],x[94]+x[277],1+x[278],-x[86]-x[87]+x[98]+x[279],

     -1+x[28],1+x[280],-1+x[86]+x[99]+x[281],-x[94]+x[282],x[87]-x[98]-x[99]+x[283],-x[94]+x[284],

     -x[86]-x[87]+x[98]+x[285],-x[94]+x[286],-1+x[86]+x[99]+x[287],-x[87]+x[288],x[87]-x[98]-x[99]+x[289],

     -1+x[29],-x[86]+x[290],1+x[291],-1+x[86]+x[87]+x[292],1+x[293],-1+x[294],-1+x[295],-1+x[296],

     1+x[298],-1+x[3],-1+x[30],1+x[301],1+x[302],1+x[303],-1+x[304],1+x[305],1+x[306],

     -1+x[86]+x[87]+x[307],1+x[308],-x[86]+x[309],1+x[31],x[94]+x[310],-x[87]+x[311],x[94]+x[312],

     -1+x[86]+x[87]+x[313],x[94]+x[314],-x[86]+x[315],-1+x[98]+x[99]+x[316],-x[87]+x[317],-x[99]+x[318],

     -1+x[319],1+x[32],-x[98]+x[320],1+x[321],1+x[322],1+x[323],-1+x[324],-1+x[325],-1+x[326],-1+x[328],

     1+x[33],-1+x[34],1+x[4],x[40]-x[94],x[41]-x[94],x[42]-x[94],-1+x[43]+x[86]+x[87],x[44]-x[86],

     x[45]-x[87],-1+x[46]+x[86]+x[87],x[47]-x[86],x[48]-x[87],1+x[5],x[53]+x[94],x[54]+x[94],x[55]+x[94],

     x[56]-x[98],x[57]-x[99],-1+x[58]+x[98]+x[99],x[59]-x[98],-1+x[6],x[60]-x[99],-1+x[61]+x[98]+x[99],

     -1+x[62],1+x[66],1+x[67],1+x[68],x[69]-x[86]-x[87]+x[98],-1+x[70]+x[86]+x[99],x[71]+x[87]-x[98]-x[99],

     x[72]-x[86]-x[87]+x[98],-1+x[73]+x[86]+x[99],x[74]+x[87]-x[98]-x[99],-1+x[75],-1+x[76],x[79]+x[94],

     x[80]+x[94],x[81]+x[94],-1+x[82]+x[86]+x[87],x[83]-x[86],x[84]-x[87],-1+x[85]+x[86]+x[87],-1+x[86]^2,

     1-x[86]-x[87]+x[86]*x[87],-x[86]-x[87]+x[98]+x[86]*x[98]+x[99]-x[87]*x[99],1-x[86]-x[99]+x[86]*x[99],

     -1+x[87]^2,-x[98]+x[87]*x[98]-x[99]+x[87]*x[99],-1+x[88],1+x[89],1+x[90],x[92]-x[94],x[93]-x[94],

     -1+x[94]^2,x[95]-x[98],x[96]-x[99],-1+x[97]+x[98]+x[99],-1+x[98]^2,1-x[98]-x[99]+x[98]*x[99],

                                                                                               -1+x[99]^2};

The following solution vector \((x[1],x[2],\ldots,x[328])\) leads to a W(46, 45) with full automorphism group of order 6.

s:={1,-1,1,-1,-1,1,1,-1,1,-1,-1,-1,1,-1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,1,-1,1,1,1,1,1,1,1,-1,1,1,-1,1,

    -1,-1,-1,1,-1,1,1,-1,1,1,1,1,1,1,-1,1,1,-1,1,1,1,-1,-1,-1,-1,-1,1,1,-1,1,1,-1,-1,1,-1,-1,-1,-1,1,

    -1,-1,-1,-1,1,1,-1,1,1,1,1,-1,-1,-1,1,-1,1,1,1,-1,-1,1,1,1,1,-1,-1,1,-1,1,-1,1,1,-1,1,1,1,1,-1,-1,

    -1,1,1,-1,1,-1,-1,1,-1,-1,1,1,1,1,-1,1,1,1,1,1,-1,-1,-1,-1,-1,-1,1,1,-1,-1,-1,1,-1,1,-1,-1,-1,1,1,

    -1,1,-1,-1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,1,-1,1,1,-1,-1,-1,1,1,1,-1,1,1,1,-1,1,-1,1,-1,-1,-1,1,-1,1,

    -1,-1,1,1,1,-1,-1,-1,1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,1,1,1,1,-1,1,1,-1,1,1,1,1,-1,-1,-1,1,1,1,1};

Appendix 4: Block Valency Matrices for n = 65

The following block valency matrices were found for the complementing permutations of type \(\sigma (4,4,4,3,3)\):

$$\displaystyle{R_{1} = \left [\begin{array}{c|cc|cc|cc||cc|cc} 0 & 8 & 0 & 8 & 0 & 8 & 0 & 4 & 0 & 4 & 0\\ \hline 1 & 6 & 4 & 2 & 4 & 4 & 4 & 1 & 2 & 2 & 2 \\ 0 & 4 & 1 & 4 & 6 & 4 & 4 & 2 & 3 & 2 & 2\\ \hline 1 & 2 & 4 & 3 & 4 & 4 & 4 & 2 & 2 & 4 & 2 \\ 0 & 4 & 6 & 4 & 4 & 4 & 4 & 2 & 2 & 2 & 0\\ \hline 1 & 4 & 4 & 4 & 4 & 2 & 4 & 4 & 2 & 1 & 2 \\ 0 & 4 & 4 & 4 & 4 & 4 & 5 & 2 & 0 & 2 & 3\\ \hline \hline 1 & 2 & 4 & 4 & 4 & 8 & 4 & 1 & 2 & 0 & 2 \\ 0 & 4 & 6 & 4 & 4 & 4 & 0 & 2 & 2 & 2 & 4\\ \hline 1 & 4 & 4 & 8 & 4 & 2 & 4 & 0 & 2 & 1 & 2 \\ 0 & 4 & 4 & 4 & 0 & 4 & 6 & 2 & 4 & 2 & 2\\ \end{array} \right ],R_{2} = \left [\begin{array}{c|cc|cc|cc||cc|cc} 0 & 8 & 0 & 8 & 0 & 8 & 0 & 4 & 0 & 4 & 0\\ \hline 1 & 3 & 4 & 4 & 2 & 4 & 4 & 2 & 2 & 2 & 4 \\ 0 & 4 & 4 & 6 & 4 & 4 & 4 & 2 & 2 & 0 & 2\\ \hline 1 & 4 & 6 & 3 & 4 & 4 & 2 & 2 & 3 & 2 & 1 \\ 0 & 2 & 4 & 4 & 4 & 6 & 4 & 1 & 2 & 3 & 2\\ \hline 1 & 4 & 4 & 4 & 6 & 3 & 4 & 2 & 0 & 2 & 2 \\ 0 & 4 & 4 & 2 & 4 & 4 & 4 & 4 & 2 & 2 & 2\\ \hline \hline 1 & 4 & 4 & 4 & 2 & 4 & 8 & 1 & 2 & 2 & 0 \\ 0 & 4 & 4 & 6 & 4 & 0 & 4 & 2 & 2 & 4 & 2\\ \hline 1 & 4 & 0 & 4 & 6 & 4 & 4 & 2 & 4 & 1 & 2 \\ 0 & 8 & 4 & 2 & 4 & 4 & 4 & 0 & 2 & 2 & 2\\ \end{array} \right ],R_{3} = \left [\begin{array}{c|cc|cc|cc||cc|cc} 0 & 8 & 0 & 8 & 0 & 8 & 0 & 4 & 0 & 4 & 0\\ \hline 1 & 4 & 4 & 4 & 3 & 4 & 5 & 0 & 2 & 3 & 2 \\ 0 & 4 & 3 & 5 & 4 & 3 & 4 & 2 & 4 & 2 & 1\\ \hline 1 & 4 & 5 & 5 & 4 & 3 & 2 & 2 & 2 & 1 & 3 \\ 0 & 3 & 4 & 4 & 2 & 6 & 5 & 2 & 2 & 1 & 3\\ \hline 1 & 4 & 3 & 3 & 6 & 5 & 4 & 2 & 2 & 1 & 1 \\ 0 & 5 & 4 & 2 & 5 & 4 & 2 & 2 & 2 & 3 & 3\\ \hline \hline 1 & 0 & 4 & 4 & 4 & 4 & 4 & 3 & 2 & 4 & 2 \\ 0 & 4 & 8 & 4 & 4 & 4 & 4 & 2 & 0 & 2 & 0\\ \hline 1 & 6 & 4 & 2 & 2 & 2 & 6 & 4 & 2 & 1 & 2 \\ 0 & 4 & 2 & 6 & 6 & 2 & 6 & 2 & 0 & 2 & 2\\ \end{array} \right ].}$$

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Szöllősi, F. (2015). The Hunt for Weighing Matrices of Small Orders. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_19

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