Processing math: 66%

Details for entry A2(7,4;3)

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Known codes

Lower bounds

Upper bounds

Bound for codes containing the lifted MRD code

(see EtzionSilberstein2012, Heinlein2018, and Kurz2019 for details)
291

Comments

HeinleinKiermaierKurzWassermann2017, Theorem 1
Let C be a set of planes in PG(6,2) mutually intersecting in at most a point. If |C|329, then the automorphism group of C is conjugate to one of the 33 subgroups of GL(7,2). The orders of of these groups are 1121324751637281192121141161. Moreover, if |C|330 then |Aut(C)|14 and if |C|334 then |Aut(C)|12.

G1,1=IC1G2,1=(1000000110000000100000011000000010000001100000001)C2G3,1=(1100000100000000110000010000000011000001000000001)C3G3,2=(1100000100000000110000010000000010000000100000001)C3G4,1=(0010100011001000110000001000101100000010100000001),(0001111011101010011010001000001011010111110000001)C2×C2G4,2=(1001010100111100110101011000101010010100100110100),(0100011111101011100111011110101001010101001101110)C2×C2G4,3=(1100010111110001100101010100101100010111100101111),(0011100101111010011001111010111011011111001011101)C2×C2G4,4=(1110011111101001011111101101110000111010110001101),(0010110001001110010100111011011011101111011010111)C2×C2G4,5=(1110101001001101010010001000000010000000100001011),(0010110001001110010100111011011011101111011010111)C2×C2G4,6=(0010000000010010000000101100010000001001101011101),(1100010110110101100101001001100010110011110100011)C2×C2G4,7=(1100000011000000100000001100000011000000110000001)C4G5,1=(0100000001000000010001111000000010000000100000001)C5G6,1=(0100110110001001111000001000100011011001000000001),(0111100111100001001100111010001100010101100000001)S3G6,2=(1101010110110000011101110000101011001101100000001),(1011010101010000001101100110100000001111100000001)S3G6,3=(1000100100111011101001011100111000001101100000001)C6G7,1=(0100000001000010100000000100000001000011000000001)C7G7,2=(0100000001000010100000000100000001000010100000001)C7G8,1=(1110100100001000010000010000111111010101000000001),(0010111100100011000010110111001110111100111001111),(0100011111001110000110100001000010010010001111011)C2×C2×C2G8,2=(1001111001101100001100111100010100101110010100011),(1010110001010000010010000111010100101101100010111),(1011101100001111100111011011000010010011000110001)C2×C2×C2G8,3=(1011000100011110000110001100011000000001111010110),(1011001100001110111101000110010100111110000000001),(0010111011100100010010111100000010011000010101100)C4G8,4=(1001111110101011100111100000101001100110110100011),(0001110110000100000100101110000010011011100110001),(1000100101111011011100100101000010001010111010010)Q8G8,5=(0010011110000000001111010101111111010110111100110),(0001110011110001000010111101101001110101001111111),(0111010111001110000110001100000010011111001010110)Q8G8,6=(1101101111110111000000101011010100100011001010010),(0001110010111100001101010000001110100011011001111),(1101101001000001000000001100000010001010110110101)D8G8,7=(0010011110000011001001001100011000000011000110001),(0100011110000111111000000111010100110000111100010),(1011001111001110000111101111000010011111000110101)C4×C2G8,8=(1001111110101011100111100000101001100110110100011),(0001010010111110011001100000000010001001000110001),(1000100101111011011100100101000010001010111010010)C4×C2G8,9=(0010011111001111011101000010010100100110110110101),(0110001010101100000101110010001110100100001101001),(0001010100011111101110111100000010011111001010010)D8G8,10=(1000000001010000010011000010010100110010001010010),(1001011110010110011001000110001110110111100010011),(0001010100011111101110111100000010011111001010010)D8G8,11=(0011100010000100101011100101010100111111010100011)C8G9,1=(1011010101110000111001100110110110001000100000001)C9G9,2=(0100010100001001100101100100110110001000000000001),(0100010101000010100100111100000101001100100000001)C3×C3G12,1=(1000011000110111111001100110000000100001110000100),(1000000110001110101011001000000010000000100000001),(1000011010111110111001100011100010010000100000010)C3C4G14,1=(0111100011000001101000100000001011010100100000001)C14G16,1=(0010100100010000010100100001101011110100100011111),(0011011101110101111100011010111101010010110010000)(C4×C2)C2

BraunKiermaierNakic2016: G_{1,1}, G_{2,1}, G_{3,1}, G_{3,2}, G_{4,7}
KiermaierKurzWassermann2018: G_{1,1}, G_{2,1}

Thomas1987: groups of order 127 are Singer cycles

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