Covering radius survey and recent results
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Title: Covering Radius Survey and Recent Results

Section 1: INTRODUCTION
Description 1: Introduce the concept of covering radius for block codes, its importance, and basic properties. Summarize the main applications and previous foundational results, along with the structure of the paper.

Section 2: The Translate Leader
Description 2: Discuss the Translate Leader concept for linear codes, including weight of a coset leader and syndrome representation. Highlight the simple criterion for identifying a translate leader.

Section 3: Perfect Codes and Quasi-perfect Codes
Description 3: Define perfect and quasi-perfect codes based on covering radius and covering radius excess, A(C). Present results related to A(C).

Section 4: UPPER BOUNDS ON COVERING RADIUS
Description 4: Outline various upper bounds on covering radius, including Redundancy, Delsarte bounds, Supercode bounds, and Norse bounds. Discuss multiple theorems providing these bounds and their proofs.

Section 5: Some Links Between t(C) and d(C)
Description 5: Explore the relationship between covering radius \( t(C) \) and minimum distance \( d(C) \). Present propositions and theorems related to maximal codes and specific classes of codes, particularly Reed-Muller codes.

Section 6: ON THE LEAST COVERING RADIUS OF (n, K) CODES
Description 6: Estimate functions \( t[n, k] \) and \( t(n, K) \) for various binary linear codes. Present lower and upper bounds on these values and discuss specific cases where exact values or tight bounds are known.

Section 7: Recent Work of Helleseth
Description 7: Summarize Helleseth's contributions related to binary cyclic codes, covering radius, and Waring's problem in \( GF(2^m) \).

Section 8: A Walsh-Transform Approach
Description 8: Describe the Walsh transform approach for calculating the covering radius, including the algorithmic complexity and computational aspects.

Section 9: Results of Wolfmann and Assmus-Pless
Description 9: Explore specific cases where the Delsarte bound on covering radius is attained. Provide related results from Wolfmann and Assmus-Pless.

Section 10: OPEN PROBLEMS
Description 10: Present open problems in the field regarding covering radius, including specific cases of codes where covering radius is not yet determined, and conjectures about code structures and behaviors.