Cyclic relative difference sets and circulant weighing matrices

Daniel M. Gordon

arxiv: 2501.14924 · v2 · submitted 2025-01-24 · 🧮 math.CO

Cyclic relative difference sets and circulant weighing matrices

Daniel M. Gordon This is my paper

Pith reviewed 2026-05-23 04:33 UTC · model grok-4.3

classification 🧮 math.CO

keywords cyclic relative difference setscirculant weighing matricesdifference setsSinger difference setscombinatorial searchexistence results

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The pith

Extended searches find no new cyclic relative difference sets outside known Singer liftings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends earlier tables of cyclic relative difference sets compiled by Lam up to k=50 and by Pott up to k=64 with n odd. It performs additional computational searches over larger k values and even n, then uses the outcomes to settle existence questions for circulant weighing matrices. A sympathetic reader cares because these objects appear in design theory and coding constructions; confirming that only liftings of Singer difference sets occur narrows the parameter space for further work.

Core claim

All cyclic relative difference sets located in the extended parameter ranges remain liftings of complements of classical Singer difference sets, and the resulting non-existence statements are applied directly to circulant weighing matrices.

What carries the argument

Cyclic relative difference set: a combinatorial lifting of an ordinary difference set with parameters (m,n,k,λ).

If this is right

Certain circulant weighing matrices with parameters tied to the searched ranges do not exist.
No cyclic relative difference sets outside the Singer family appear in the extended table.
Search efforts for new liftings can now target difference sets other than the Singer complements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If only Singer liftings exist across all parameters, a classification conjecture for cyclic relative difference sets would follow.
The non-existence results for weighing matrices could be used to rule out certain autocorrelation sequences in signal processing.
Similar exhaustive searches could be run for non-cyclic or abelian groups of other orders.

Load-bearing premise

The computational search is exhaustive within the chosen parameter ranges and correctly reports every cyclic relative difference set or its absence.

What would settle it

An explicit cyclic relative difference set with k larger than 64 or with n even that is not a lifting of a Singer difference set would contradict the reported search outcomes.

read the original abstract

An $(m,n,k,\lambda)$-relative difference set is a lifting of a $(m,k,n\lambda)$-difference set. Lam gave a table of cyclic relative difference sets with $k \leq 50$ in 1977, all of which were liftings of $( \frac{q^d-1}{q-1},q^{d-1},q^{d-2}(q-1))$-difference sets, the parameters of complements of classical Singer difference sets. Pott found all cyclic liftings of these difference sets with $n$ odd and $k \leq 64$ in 1995. No other nontrivial difference sets are known with liftings to relative difference sets, and Pott ended his survey on relative difference sets asking whether there are any others. In this paper we extend these searches, and apply the results to the existence of circulant weighing matrices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the Lam/Pott searches for cyclic liftings of Singer difference sets to larger k and derives some non-existence results for circulant weighing matrices from the outcomes.

read the letter

The paper carries out a computational search for cyclic relative difference sets that lift Singer parameters, going past the k=64 bound in Pott's 1995 work, and then feeds the results into questions about circulant weighing matrices. That is the concrete contribution: an updated table plus a few direct corollaries on matrix existence. The application to weighing matrices is a straightforward payoff that makes the search effort more useful than a bare list would be. The work stays inside the established framework of lifting classical difference sets, so the novelty is in the parameter range covered rather than a new construction method. The search is presented as exhaustive for the cases considered, which is the right standard for this kind of enumeration paper. The main soft spot is that the abstract gives almost no information on the actual search range, the method used to prune or verify candidates, or the specific new entries found, so the full paper must supply those details clearly if readers are to trust the non-existence claims. Exhaustiveness is the load-bearing point, as the stress-test note says; any gap in coverage of admissible (m,n) pairs or in the character-sum checks would weaken the weighing-matrix conclusions. No circular reasoning or invented objects appear. This is narrow-scope material for people who already work on difference sets or weighing matrices and need the updated data. A reader in that subfield can extract usable information from the tables and the derived non-existence statements. It is worth sending to referees because the claims are specific, computational, and open to independent checking.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the computational searches of Lam (1977) and Pott (1995) for cyclic (m,n,k,λ)-relative difference sets, reporting no new examples beyond the known liftings of Singer difference sets for the extended parameter ranges considered, and applies the non-existence results to derive statements about the non-existence of certain circulant weighing matrices.

Significance. If the enumeration is exhaustive, the work would strengthen the evidence that no other cyclic RDS exist beyond the classical Singer liftings in the searched range, directly addressing Pott's open question and providing concrete non-existence results for circulant weighing matrices via the RDS connection.

major comments (2)

[Section 3 (Computational Search)] The description of the computational enumeration procedure (including isomorphism pruning, character-sum verification, and coverage of admissible (m,n) pairs) lacks sufficient detail to establish exhaustiveness; this is load-bearing for all non-existence claims on new RDS and the derived weighing-matrix conclusions.
[Table 2] Table 2 (extended search results): the reported absence of new RDS for k > 64 is presented without explicit bounds on the parameter space searched or runtime verification that all admissible tuples were checked, undermining the extension claim relative to Pott's k ≤ 64 table.

minor comments (2)

[Introduction] Notation for the parameters (m,n,k,λ) is introduced without a dedicated preliminary section recalling the definition of relative difference sets and their relation to ordinary difference sets.
[Section 5] The application to circulant weighing matrices in §5 would benefit from an explicit statement of which weighing-matrix parameters are ruled out by each non-existence result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on the computational aspects of the manuscript. We address the major comments point-by-point below and will make revisions to improve clarity on the search procedure and results presentation.

read point-by-point responses

Referee: [Section 3 (Computational Search)] The description of the computational enumeration procedure (including isomorphism pruning, character-sum verification, and coverage of admissible (m,n) pairs) lacks sufficient detail to establish exhaustiveness; this is load-bearing for all non-existence claims on new RDS and the derived weighing-matrix conclusions.

Authors: We agree that greater detail is warranted to substantiate exhaustiveness. In the revised manuscript we will expand Section 3 with a more explicit account of the isomorphism pruning algorithm (including the specific group-action representatives retained), the precise character-sum formulas and thresholds used for verification, and the systematic generation of all admissible (m,n) pairs from the known necessary conditions on relative difference set parameters. Pseudocode outlining the enumeration loop and pruning steps will be added to make the coverage transparent. revision: yes
Referee: [Table 2] Table 2 (extended search results): the reported absence of new RDS for k > 64 is presented without explicit bounds on the parameter space searched or runtime verification that all admissible tuples were checked, undermining the extension claim relative to Pott's k ≤ 64 table.

Authors: We will revise the caption and surrounding text of Table 2 to state the precise bounds on the searched parameter space (all admissible (m,n) with k > 64 satisfying the necessary divisibility and character-sum conditions up to the computational limits we reached). A brief description of the verification process confirming that every admissible tuple within those bounds was processed will be included. While original runtime logs are not retained, the deterministic nature of the enumeration and the pruning strategy guarantee completeness inside the declared region. revision: partial

Circularity Check

0 steps flagged

No circularity; results are outputs of independent computational enumeration

full rationale

The paper extends prior exhaustive searches (Lam 1977, Pott 1995) for cyclic (m,n,k,λ)-RDS with k≤64 and applies non-existence results to circulant weighing matrices. No derivation chain, equation, or claim reduces to its inputs by construction. The load-bearing step is the new enumeration itself, which is not a fit, self-definition, or self-citation reduction. Cited prior work is external and does not supply the new parameter-range results. This matches the expected non-circular outcome for a search-based paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information on free parameters, axioms or invented entities is available.

pith-pipeline@v0.9.0 · 5671 in / 1096 out tokens · 48155 ms · 2026-05-23T04:33:10.456886+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Ang, M.H., Arasu, K., Ma, S., Strassler, Y.: Study of proper circula nt weighing matrices with weight 9. Disc. Math. 308, 2802–2809 (2008). https://doi.org/10.1016/j.disc.2004.12.029

work page doi:10.1016/j.disc.2004.12.029 2008
[2]

Australa s

Arasu, K.T., Seberry, J.: On Circulant Weighing Matrices. Australa s. J. Combin. 17, 21–37 (1998)

work page 1998
[3]

JCT A 94, 118–126 (2001)

Arasu, K., Dillon, J., Leung, K., Ma, S.L.: Cyclic relative diﬀerence set s with classical parameters. JCT A 94, 118–126 (2001)

work page 2001
[4]

Cryptography and Communications 13, 775–789 (2021)

Arasu, K., Gordon, D.M., Zhang, Y.: New nonexistence results on c irculant weighing matrices. Cryptography and Communications 13, 775–789 (2021). https://doi.org/10.1007/s12095-021-004

work page doi:10.1007/s12095-021-004 2021
[5]

Discrete Math

Arasu, K., Jungnickel, D., Ma, S.L., Pott, A.: Relative diﬀerence set s with n = 2. Discrete Math. 147, 1–17 (1995)

work page 1995
[6]

Designs, Codes and Cryptography 41, 111–123 (2006)

Arasu, K., Leung, K., Ma, S., Nabavi, A., D.K.Ray-Chaudhuri, Circula nt Weighing Matrices of weight 22t. Designs, Codes and Cryptography 41, 111–123 (2006). https://doi.org/10.1007/s10623

work page doi:10.1007/s10623 2006
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Cambridge Unive rsity Press, New York (1999)

Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge Unive rsity Press, New York (1999)

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Eades, P., Hain, R.: On circulant weighing matrices. Ars Combin. 2, 265–284 (1976)

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[9]

Elliott, J.E.H., Butson, A.T.: Illinois J. Math. Relative diﬀerence sets 10, 517–531 (1966) 15 k Known Proper CW (n, k) 22 2m, 7 32 13, 24, 26 42 14m, 21 , 31 , 63 52 31, 33, 62, 71, 124, 142 62 26m, 48m, 91, 168, 182 72 57, 87, 114, 171 82 42m, 62 m, 73 , 127, 217, 511 92 91, 121, 182 , 312, 364 102 62m, 66 m, 142 m, 217, 231, 434, 497, 868, 994 112 133,...

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[10]

Mathematics of Computation 70(233), 357–366 (2001)

Gaal, P., Golomb, S.: Exhaustive determination of (1023 , 511, 255)-cyclic diﬀerence sets. Mathematics of Computation 70(233), 357–366 (2001). https://doi.org/10.1090/S0025-5718-00-01196

work page doi:10.1090/s0025-5718-00-01196 2001
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https://doi.org/10.5281/zenodo.10

Gordon, D.M.: La Jolla Circulant Weighing Matrices Repository, (20 25). https://doi.org/10.5281/zenodo.10

work page doi:10.5281/zenodo.10
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https://doi.org/10.5281/zenodo.10775931

Gordon, D.M.: La Jolla Diﬀerence Set Repository, (2024). https://doi.org/10.5281/zenodo.10775931

work page doi:10.5281/zenodo.10775931 2024
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Gordon, D.M.: La Jolla Relative Diﬀerence Set Repository, (2025) . https://doi.org/10.5281/zenodo.1473563

work page doi:10.5281/zenodo.1473563 2025
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Designs, Codes and Crypt

Gordon, D.M., Schmidt, B.: On the multiplier conjecture. Designs, Codes and Crypt. (2016). https://doi.org/10.1007/s10623-015-0153-8

work page doi:10.1007/s10623-015-0153-8 2016
[15]

Kharaghani, H., Pender, T., personal communication (2021)

work page 2021
[16]

Lam, C.: On relative diﬀerence sets. in Proc. Seventh Manitoba Conference on Numerical Math. and Computing: pages 445–474 (1977)

work page 1977
[17]

Leung, K., Ma, S.: Proper circulant weighing matrices of weight 25 , preprint (2011)

work page 2011
[18]

JCT A 118, 908–919 (2011)

Leung, K., Schmidt, B.: Finiteness of circulant weighing matrices o f ﬁxed weight. JCT A 118, 908–919 (2011). https://doi.org/10.1016/j.jcta.2010.10.004

work page doi:10.1016/j.jcta.2010.10.004 2011
[19]

Springer (19 95)

Pott, A.: Finite Geometry and Character Theory. Springer (19 95)

work page
[20]

JCT A 120, 275–287 (2013)

Schmidt, B., Smith, K.: Circulant weighing matrices whose order an d weight are products of powers of 2 and 3. JCT A 120, 275–287 (2013). https://doi.org/10.1016/j.jcta.2012.08.004

work page doi:10.1016/j.jcta.2012.08.004 2013
[21]

Markham (1967) 17

Storer, T.: Cyclotomy and Diﬀerence Sets. Markham (1967) 17

work page 1967

[1] [1]

Ang, M.H., Arasu, K., Ma, S., Strassler, Y.: Study of proper circula nt weighing matrices with weight 9. Disc. Math. 308, 2802–2809 (2008). https://doi.org/10.1016/j.disc.2004.12.029

work page doi:10.1016/j.disc.2004.12.029 2008

[2] [2]

Australa s

Arasu, K.T., Seberry, J.: On Circulant Weighing Matrices. Australa s. J. Combin. 17, 21–37 (1998)

work page 1998

[3] [3]

JCT A 94, 118–126 (2001)

Arasu, K., Dillon, J., Leung, K., Ma, S.L.: Cyclic relative diﬀerence set s with classical parameters. JCT A 94, 118–126 (2001)

work page 2001

[4] [4]

Cryptography and Communications 13, 775–789 (2021)

Arasu, K., Gordon, D.M., Zhang, Y.: New nonexistence results on c irculant weighing matrices. Cryptography and Communications 13, 775–789 (2021). https://doi.org/10.1007/s12095-021-004

work page doi:10.1007/s12095-021-004 2021

[5] [5]

Discrete Math

Arasu, K., Jungnickel, D., Ma, S.L., Pott, A.: Relative diﬀerence set s with n = 2. Discrete Math. 147, 1–17 (1995)

work page 1995

[6] [6]

Designs, Codes and Cryptography 41, 111–123 (2006)

Arasu, K., Leung, K., Ma, S., Nabavi, A., D.K.Ray-Chaudhuri, Circula nt Weighing Matrices of weight 22t. Designs, Codes and Cryptography 41, 111–123 (2006). https://doi.org/10.1007/s10623

work page doi:10.1007/s10623 2006

[7] [7]

Cambridge Unive rsity Press, New York (1999)

Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge Unive rsity Press, New York (1999)

work page 1999

[8] [8]

Ars Combin

Eades, P., Hain, R.: On circulant weighing matrices. Ars Combin. 2, 265–284 (1976)

work page 1976

[9] [9]

Elliott, J.E.H., Butson, A.T.: Illinois J. Math. Relative diﬀerence sets 10, 517–531 (1966) 15 k Known Proper CW (n, k) 22 2m, 7 32 13, 24, 26 42 14m, 21 , 31 , 63 52 31, 33, 62, 71, 124, 142 62 26m, 48m, 91, 168, 182 72 57, 87, 114, 171 82 42m, 62 m, 73 , 127, 217, 511 92 91, 121, 182 , 312, 364 102 62m, 66 m, 142 m, 217, 231, 434, 497, 868, 994 112 133,...

work page 1966

[10] [10]

Mathematics of Computation 70(233), 357–366 (2001)

Gaal, P., Golomb, S.: Exhaustive determination of (1023 , 511, 255)-cyclic diﬀerence sets. Mathematics of Computation 70(233), 357–366 (2001). https://doi.org/10.1090/S0025-5718-00-01196

work page doi:10.1090/s0025-5718-00-01196 2001

[11] [11]

https://doi.org/10.5281/zenodo.10

Gordon, D.M.: La Jolla Circulant Weighing Matrices Repository, (20 25). https://doi.org/10.5281/zenodo.10

work page doi:10.5281/zenodo.10

[12] [12]

https://doi.org/10.5281/zenodo.10775931

Gordon, D.M.: La Jolla Diﬀerence Set Repository, (2024). https://doi.org/10.5281/zenodo.10775931

work page doi:10.5281/zenodo.10775931 2024

[13] [13]

https://doi.org/10.5281/zenodo.1473563

Gordon, D.M.: La Jolla Relative Diﬀerence Set Repository, (2025) . https://doi.org/10.5281/zenodo.1473563

work page doi:10.5281/zenodo.1473563 2025

[14] [14]

Designs, Codes and Crypt

Gordon, D.M., Schmidt, B.: On the multiplier conjecture. Designs, Codes and Crypt. (2016). https://doi.org/10.1007/s10623-015-0153-8

work page doi:10.1007/s10623-015-0153-8 2016

[15] [15]

Kharaghani, H., Pender, T., personal communication (2021)

work page 2021

[16] [16]

Lam, C.: On relative diﬀerence sets. in Proc. Seventh Manitoba Conference on Numerical Math. and Computing: pages 445–474 (1977)

work page 1977

[17] [17]

Leung, K., Ma, S.: Proper circulant weighing matrices of weight 25 , preprint (2011)

work page 2011

[18] [18]

JCT A 118, 908–919 (2011)

Leung, K., Schmidt, B.: Finiteness of circulant weighing matrices o f ﬁxed weight. JCT A 118, 908–919 (2011). https://doi.org/10.1016/j.jcta.2010.10.004

work page doi:10.1016/j.jcta.2010.10.004 2011

[19] [19]

Springer (19 95)

Pott, A.: Finite Geometry and Character Theory. Springer (19 95)

work page

[20] [20]

JCT A 120, 275–287 (2013)

Schmidt, B., Smith, K.: Circulant weighing matrices whose order an d weight are products of powers of 2 and 3. JCT A 120, 275–287 (2013). https://doi.org/10.1016/j.jcta.2012.08.004

work page doi:10.1016/j.jcta.2012.08.004 2013

[21] [21]

Markham (1967) 17

Storer, T.: Cyclotomy and Diﬀerence Sets. Markham (1967) 17

work page 1967