(* Uniform rank 2 matroid on 5 elements *)
W52 = {{0,w12,w13,w14,w15},{w12,0,w23,w24,w25},{w13,w23,0,w34,w35},{w14,w24,w34,0,w45},{w15,w25,w35,w45,0}}; (* Matrix of scalings *)
det52 = Det[W52]; (* Determinant of W *)
M52 = Minors[W52,4]; (* 4-minors of W *)
Simplify[FindInstance[det52==0&&Table[M52[[i]][[i]],{i,5}]==0&&w12≠0&&w13≠0&&w14≠0&&w15≠0&&w23≠0&&w24≠0&&w25≠0&&w34≠0&&w35≠0&&w45≠0,{w12,w13,w14,w15,w23,w24,w25,w34,w35,w45},Complexes]](* find ML degree 6 scaling *)

(* Uniform rank 2 matroid on 6 elements *)
W62 = {{0,w12,w13,w14,w15,w16},{w12,0,w23,w24,w25,w26},{w13,w23,0,w34,w35,w36},{w14,w24,w34,0,w45,w46},{w15,w25,w35,w45,0,w56},{w16,w26,w36,w46,w56,0}}; (* Matrix of scalings *)
det62 = Det[W62]; (* Determinant of W *)
M624 = Minors[W62,4]; (* 4-minors of W *)
M625 = Minors[W62,5]; (* 5-minors of W *)
R62 = ReplaceAll[{w12->-1,w13->-1,w14->-1,w15->-1,w23->-1,w24->-1}][{Table[M624[[i]][[i]],{i,15}]}]; (* fix some scalings *)
FindInstance[Table[R62[[1]][[i]],{i,15}]==0&&w16!=0&&w25!=0&&w26!=0&&w34!=0&&w35!=0&&w36!=0&&w45!=0&&w46!=0&&w56!=0,{w16,w25,w26,w34,w35,w36,w45,w46,w56},Complexes,1](* find ML degree 10 scaling *) 
Simplify[ReplaceAll[{w12->-1,w13->-1,w14->-1,w15->-1,w16->(1/8) (3-12 I (Sqrt[3])),w23->-1,w24->-1,w25->8,w26->-12,w34->-4,w35->7-4 I (Sqrt[2]),w36->(1/16) (-45-21 (7-4 I (Sqrt[2]))+9 (3-12 I (Sqrt[3]))-(7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w45->7+4 I (Sqrt[2]),w46->(1/16) (-339+21 (7-4 I (Sqrt[2]))-5 (3-12 I (Sqrt[3]))+(7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w56->9}][{Table[M624[[i]][[i]],{i,15}],Table[M625[[i]][[i]],{i,6}],det62  }]] (* check if all principal minors vanish *)

(* Uniform rank 2 matroid on 7 elements *)
W72 = {{0,w12,w13,w14,w15,w16,w17},{w12,0,w23,w24,w25,w26,w27},{w13,w23,0,w34,w35,w36,w37},{w14,w24,w34,0,w45,w46,w47},{w15,w25,w35,w45,0,w56,w57},{w16,w26,w36,w46,w56,0,w67},{w17,w27,w37,w47,w57,w67,0}}; (* Matrix of scalings *)
det72 = Det[W72] ;(* Determinant of W *)
M724 = Minors[W72,4] ;(* 4-minors of W *)
M725 = Minors[W72,5]; (* 5-minors of W *)
M726 = Minors[W72,6]; (* 6-minors of W *)
R72 = ReplaceAll[{w12->-1,w13->-1,w14->-1,w15->-1,w23->-1,w24->-1,w25->8,w26->-12,w34->-4,w56->9}][{Table[M724[[i]][[i]],{i,35}]}];   (* fix some scalings *)
FindInstance[Table[R72[[1]][[i]],{i,35}]==0&&w16!=0&&w17!=0&&w27!=0&&w35!=0&&w36!=0&&w37!=0&&w45!=0&&w46!=0&&w47!=0&&w57!=0&&w67!=0,{w16,w17,w27,w35,w36,w37,w45,w46,w47,w57,w67},Complexes,1] (* find ML degree 15 scaling *) 
Simplify[ReplaceAll[{w12->-1,w13->-1,w14->-1,w15->-1,w16->(1/8) (3-12 I (Sqrt[3])),w17->-1,w23->-1,w24->-1,w25->8,w26->-12,w27->1,w34->-4,w35->7-4 I (Sqrt[2]),w36->(1/16) (-45-21 (7-4 I (Sqrt[2]))+9 (3-12 I (Sqrt[3]))-(7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w37->-2 I,w45->7+4 I (Sqrt[2]),w46->(1/16) (-339+21 (7-4 I (Sqrt[2]))-5 (3-12 I (Sqrt[3]))+(7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w47->2 I,w56->9,w57->(1/2) ((18+14 I)-2 I (7-4 I (Sqrt[2]))),w67->(1/32) ((-384-294 I)+42 I (7-4 I (Sqrt[2]))-(4+14 I) (3-12 I (Sqrt[3]))+2 I (7-4 I (Sqrt[2])) (3-12 I (Sqrt[3])))}][{Table[M724[[i]][[i]],{i,35}],Table[M725[[i]][[i]],{i,21}],Table[M726[[i]][[i]],{i,7}],det72}]](* check if all principal minors vanish *)

(* Uniform rank 2 matroid on 8 elements *)
W82 = {{0,w12,w13,w14,w15,w16,w17,w18},{w12,0,w23,w24,w25,w26,w27,w28},{w13,w23,0,w34,w35,w36,w37,w38},{w14,w24,w34,0,w45,w46,w47,w48},{w15,w25,w35,w45,0,w56,w57,w58},{w16,w26,w36,w46,w56,0,w67,w68},{w17,w27,w37,w47,w57,w67,0,w78},{w18,w28,w38,w48,w58,w68,w78,0}};(* Matrix of scalings *)
det82 = Det[W82]; (* Determinant of W *)
M824 = Minors[W82,4]; (* 4-minors of W *)
M825 = Minors[W82,5]; (* 5-minors of W *)
M826 = Minors[W82,6]; (* 6-minors of W *)
M827 = Minors[W82,7]; (* 7-minors of W *)
R82 = ReplaceAll[{w12->-1,w13->-1,w14->-1,w15->-1,w17->-1,w23->-1,w24->-1,w25->8,w26->-12,w27->1,w34->-4,w56->9}][{Table[M824[[i]][[i]],{i,70}]}];(* fix some scalings *)
FindInstance[Table[R82[[1]][[i]],{i,70}]==0&&w16!=0&&w18!=0&&w28!=0&&w35!=0&&w36!=0&&w37!=0&&w38!=0&&w45!=0&&w46!=0&&w47!=0&&w48!=0&&w57!=0&&w58!=0&&w67!=0&&w68!=0&&w78!=0,{w16,w18,w28,w35,w36,w37,w38,w45,w46,w47,w48,w57,w58,w67,w68,w78},Complexes,1](* find ML degree 21 scaling *) 
Simplify[ReplaceAll[{w12->-1,w13->-1,w14->-1,w15->-1,w16->(1/8) (3-12 I (Sqrt[3])),w17->-1,w18->-4,w23->-1,w24->-1,w25->8,w26->-12,w27->1,w28->1,w34->-4,w35->7-4 I (Sqrt[2]),w36->(1/16) (-45-21 (7-4 I (Sqrt[2]))+9 (3-12 I (Sqrt[3]))-(7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w37->-2 I,w38->-3-4 I,w45->7+4 I (Sqrt[2]),w46->(1/16) (-339+21 (7-4 I (Sqrt[2]))-5 (3-12 I (Sqrt[3]))+(7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w47->2 I,w48->-3+4 I,w56->9,w57->(1/2) ((18+14 I)-2 I (7-4 I (Sqrt[2]))),w58->(1/2) ((66+28 I)-4 I (7-4 I (Sqrt[2]))),w67->(1/32) ((-384-294 I)+42 I (7-4 I (Sqrt[2]))-(4+14 I) (3-12 I (Sqrt[3]))+2 I (7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w68->(1/32) ((-1536-588 I)+84 I (7-4 I (Sqrt[2]))-(4+28 I) (3-12 I (Sqrt[3]))+4 I (7-4 I (Sqrt[2])) (3-12 I (Sqrt[3]))),w78->1}][{Table[M824[[i]][[i]],{i,70}],Table[M825[[i]][[i]],{i,56}],Table[M826[[i]][[i]],{i,28}], Table[M827[[i]][[i]],{i,8}],det82}]](* check if all principal minors vanish *)