Abstract
Let \(q=2^m\). The projective general linear group \({\mathrm {PGL}}(2,q)\) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over \({\mathrm {GF}}(2^h)\) that are invariant under \({\mathrm {PGL}}(2,q)\) are trivial codes: the repetition code, the whole space \({\mathrm {GF}}(2^h)^{2^m+1}\), and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all \(3-(q+1,k,\lambda )\) designs that are invariant under \({\mathrm {PGL}}(2,q)\) are determined. The second objective is to present two infinite families of cyclic codes over \({\mathrm {GF}}(2^m)\) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under \({\mathrm {PGL}}(2,q)\), therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters \([q+1,q-3,4]_q\), where \(q=2^m\), and \(m\ge 4\) is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-\((q+1,4,2)\) design. A code from the second family has parameters \([q+1,4,q-4]_q\), \(q=2^m\), \(m\ge 4\) even, and the minimum weight codewords support a 3-\((q +1,q-4,(q-4)(q-5)(q-6)/60)\) design, whose complementary 3-\((q +1, 5, 1)\) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over \({\mathrm {GF}}(q)\) that can support a 3-\((q +1,q-4,(q-4)(q-5)(q-6)/60)\) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.
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References
Assmus Jr. E.F., Mattson Jr. H.F.: New 5-designs. J. Comb. Theory 6, 122–151 (1969).
Beth T., Jungnickel D., Lenz H.: Design Theory. Cambridge University Press, Cambridge (1999).
Bosma W., Cannon J.: Handbook of Magma Functions. University of Sydney, School of Mathematics and Statistics, Sydney (1999).
Colbourn C.J., Dinitz J.F.: Handbook of Combinatorial Designs, 2nd edn. Chapman & Hall/CRC, Boca Raton (2007).
Dickson L.E.: Linear Groups: With an Exposition of the Galois Field Theory. Teubner, Leipzig (1901).
Delsarte P.: On subfield subcodes of modified Reed-Solomon codes. IEEE Trans. Inform. Theory 21(5), 575–576 (1975).
Ding C.: Designs from Linear Codes. World Scientific, Singapore (2018).
Ding C., Tang C.: Infinite families of near MDS codes holding \(t\)-designs. IEEE Trans. Inform. Theory 66(9), 5419–5428 (2020).
Du X., Wang R., Fan C.: Infinite families of \(2\)-designs from a class of cyclic codes. J. Comb. Des. 28(3), 157–170 (2020).
Giorgetti M., Previtali A.: Galois invariance, trace codes and subfield subcodes. Finite Fields Appl. 16(2), 96–99 (2010).
Huber M.: The classification of flag-transitive Steiner 3-designs. Adv. Geom. 5(2), 195–221 (2005).
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).
Hughes D.R.: On \(t\)-designs and groups. Am. J. Math. 87(4), 761–778 (1965).
Jungnickel D., Magliveras S.S., Tonchev V.D., Wassermann A.: The classification of Steiner triple systems on 27 points with 3-rank 24. Des. Codes Cryptogr. 87, 831–839 (2019).
Jungnickel D., Tonchev V.D.: New invariants for incidence structures. Des. Codes Cryptogr. 68, 163–177 (2013).
Jungnickel D., Tonchev V.D.: Counting Steiner triple systems with classical parameters and prescribed rank. J. Comb. Theory Ser. A 162, 10–33 (2019).
Passman D.S.: Permutation Groups. Benjamin, New York (1968).
Shi M., Xu L., Krotov D.S.: The number of the non-full-rank Steiner triple systems. J. Comb. Des. 27(10), 571–585 (2019).
Tang C.: Infinite families of \(3\)-designs from APN functions. J. Comb. Des. 28(2), 97–117 (2020).
Tang C., Ding C.: An infinite family of linear codes supporting \(4\)-designs. IEEE Trans. Inform. Theory 67(1), 244–254 (2021).
Tonchev V.D.: Linear perfect codes and a characterization of the classical designs. Des. Codes Cryptogr. 17, 121–128 (1999).
Tonchev V.D.: Codes. In: Colbourn C.J., Dinitz J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn, pp. 677–701. CRC Press, New York (2007).
Witt E.: Über Steinersche Systeme. Abh. Math. Sem. Hamburg 12, 265–275 (1938).
Xiang C., Ling X., Wang Q.: Combinatorial \(t\)-designs from quadratic functions. Des. Codes Cryptogr. 88(3), 553–565 (2020).
Zinoviev D.V.: The number of Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+2\) over \(\mathbb{F}_2\). Discrete Math. 339, 2727–2736 (2016).
Zinoviev V.A., Zinoviev D.V.: Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+1\) over \(\mathbb{F}_2\). Probl. Inf. Transm. 48, 102–126 (2012).
Zinoviev V.A., Zinoviev D.V.: Structure of Steiner triple systems \(S(2^m - 1, 3, 2)\) of rank \(2^m -m+2\) over \(\mathbb{F}_2\). Probl. Inf. Transm. 49, 232–248 (2013).
Zinoviev V.A., Zinoviev D.V.: Remark on Steiner triple systems \( S(2^m - 1, 3, 2)\) of rank \(2^m -m+1\) over \( {{\mathbb{F}}_2}\) published in Probl. Peredachi Inf., 2012, no. 2. Probl. Inf. Transm. 49, 107–111 (2013).
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C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16300418. C. Tang’s research was supported by The National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds).
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Ding, C., Tang, C. & Tonchev, V.D. The projective general linear group \({\mathrm {PGL}}(2,2^m)\) and linear codes of length \(2^m+1\). Des. Codes Cryptogr. 89, 1713–1734 (2021). https://doi.org/10.1007/s10623-021-00888-2
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DOI: https://doi.org/10.1007/s10623-021-00888-2