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### Separating Codes: Constructions and Bounds [chapter]

Gérard Cohen, Hans Georg Schaathun
2004 Lecture Notes in Computer Science
Separating codes play their role in making the fingerprinting scheme secure agains coalitions of pirates. We provide here better bounds, constructions and generalizations for these codes.  ...  Separating codes, initially introduced to test automaton, have revived lately in the study of fingerprinting codes, which are used for copyright protection.  ...  There is a constructive asymptotic family of binary (2, 1)-separating codes with rate R = 0.1845.  ...

### Bounds and Constructions for 3-Strongly Separable Codes with Length 3 [article]

Xuli Zhang, Jing Jiang, Minquan Cheng
2017 arXiv   pre-print
Codes and Cryptogr.79:303-318, 2016) can be also used to construct anti-collusion codes.  ...  As separable code (SC, IEEE Trans Inf Theory 57:4843-4851, 2011) and frameproof code (FPC, IEEE Trans Inf Theory 44:1897-1905, 1998) do in multimedia fingerprinting, strongly separable code (SSC, Des.  ...  FA380009 and CA380021) and 2014GXNSFDA(No. 118001).  ...

### Bounds and Constructions for 3-Separable Codes with Length 3 [article]

Minquan Cheng, Jing Jiang, Haiyan Li, Ying Miao, Xiaohu Tang
2015 arXiv   pre-print
In this paper, an upper bound on M(3,3,q) is derived by considering an optimization problem related to a partial Latin square, and then two constructions for 3-SC(3,M,q)s are provided by means of perfect  ...  C is a t-separable code if for any two distinct C_1, C_2 ⊆C with |C_1| < t, |C_2| < t, we always have desc(C_1) ≠ desc(C_2). Let M(t,n,q) denote the maximal possible size of such a separable code.  ...  Cheng and Miao [8] showed that long-length separable codes can be constructed by concatenating short-length separable codes.  ...

### Separating hash families: A Johnson-type bound and new constructions [article]

Chong Shangguan, Gennian Ge
2016 arXiv   pre-print
In this paper, using tools from graph theory and additive number theory, we solve several open problems and conjectures concerning bounds and constructions for separating hash families.  ...  Secondly, we present a construction for an infinite class of perfect hash families. It is based on the Hamming graphs in coding theory and generalizes many constructions that appeared before.  ...  We call this recursive bound a "Johnson-type bound" due to its similarity with the traditional recursive Johnson bound in coding theory.  ...

### Constructions and Bounds for Mixed-Dimension Subspace Codes [article]

Thomas Honold, Michael Kiermaier, Sascha Kurz
2018 arXiv   pre-print
For d=v-2 we present some improved bounds and determine A_q(5,3)=2q^3+2 (all q), A_2(7,5)=34.  ...  We also provide an exposition of the known determination of A_q(v,2), and a table with exact results and bounds for the numbers A_2(v,d), v≤ 7.  ...  Acknowledgments The authors are grateful to Tuvi Etzion for comments regarding the upper bound in Theorem 4.1 and to two anonymous reviewers for valuable comments and corrections.  ...

### Construction and bounds for subspace codes [article]

Sascha Kurz
2021 arXiv   pre-print
They have e.g. applications in random linear network coding, distributed storage, and cryptography. In this chapter we survey known constructions and upper bounds for subspace codes.  ...  Subspace codes are the q-analog of binary block codes in the Hamming metric. Here the codewords are vector spaces over a finite field.  ...  57 5 Constructions for constant dimension codes 5.4 Constructions for 𝑑 -packings of CDCs and RMCs As already mentioned, we can separate the problem of the choice of the  ...

### Matroid Theory and Storage Codes: Bounds and Constructions [article]

Ragnar Freij-Hollanti, Camilla Hollanti, Thomas Westerbäck
2017 arXiv   pre-print
Recent research on distributed storage systems (DSSs) has revealed interesting connections between matroid theory and locally repairable codes (LRCs).  ...  The chapter embeds all the essential features of LRCs, namely locality, availability, and hierarchy alongside with related generalised Singleton bounds.  ...  The first example of a storage code, and the motivating example behind the notion of locality, is the notion of a maximum distance separable (MDS) code.  ...

### Bounds and Constructions for Multi-Symbol Duplication Error Correcting Codes [article]

Andreas Lenz, Niklas Jünger, Antonia Wachter-Zeh
2018 arXiv   pre-print
Finally, we discuss the asymptotic behavior of the derived codes and bounds, which yields insights about the tandem duplication channel.  ...  Based on the fact that a code correcting insertions of t zero-blocks can be used to correct t tandem duplications, we construct codes for tandem duplication errors.  ...  We have further found two constructions can correct Fig. 1 .Fig. 2 . 12 Redundancy of constructions and bounds (p = 2, t = 1, ℓ = 3) Redundancy of constructions and bounds (p = 2, t = 3, ℓ = 3) This  ...

### Bounds and Constructions for Multireceiver Authentication Codes [chapter]

R. Safavi-Naini, H. Wang
1998 Lecture Notes in Computer Science
In this paper, we give a formal definition of multireceiver authentication codes, derive information theoretic and combinatorial lower bounds on their performance and give new efficient constructions for  ...  such codes, our constructions are based on the linear error-correcting codes.  ...  Using maximum distance separable codes in the first construction, and special values for parameters in the second, results in new optimal MRA-codes that satisfy lower bounds on the size of keys and the  ...

### Refined Coding Bounds and Code Constructions for Coherent Network Error Correction

Shenghao Yang, Raymond W. Yeung, Chi Kin Ngai
2011 IEEE Transactions on Information Theory
The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound.  ...  With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for coherent network  ...  The relation between network coding and maximum distance separation (MDS) codes in classical algebraic coding [14] was clarified in [15] .  ...

### Bounds and Constructions of Codes with Multiple Localities [article]

Alexander Zeh, Eitan Yaakobi
2016 arXiv   pre-print
This paper studies bounds and constructions of locally repairable codes (LRCs) with multiple localities so-called multiple-locality LRCs (ML-LRCs).  ...  Furthermore, we construct Singleton-optimal ML-LRCs as well as codes that achieve the extended alphabet-dependent bound.  ...  For r ≥ k Thm. 2 coincides with the classical (locality-unaware) Singleton bound and then a Singleton-optimal code is called maximum distance separable (MDS).  ...

### New LMRD bounds for constant dimension codes and improved constructions [article]

Daniel Heinlein
2018 arXiv   pre-print
We generalize upper bounds for constant dimension codes containing a lifted maximum rank distance code first studied by Etzion and Silberstein. The proof allows to construct several improved codes.  ...  A prominent code construction uses maximum rank distance (MRD) codes.  ...  [2, 23] often surpasses the LMRD bound and is therefore stronger than all constructions that include an LMRD as subset.  ...

### Bounds and constructions of codes with multiple localities

Alexander Zeh, Eitan Yaakobi
2016 2016 IEEE International Symposium on Information Theory (ISIT)
This paper studies bounds and constructions of locally repairable codes (LRCs) with multiple localities so-called multiplelocality LRCs (ML-LRCs).  ...  Furthermore, we construct Singleton-optimal ML-LRCs as well as codes that achieve the extended alphabet-dependent bound.  ...  For r ≥ k Thm. 2 coincides with the classical (locality-unaware) Singleton bound and then a Singleton-optimal code is called maximum distance separable (MDS).  ...

### Constacyclic symbol-pair codes: lower bounds and optimal constructions [article]

Bocong Chen, Liren Lin, Hongwei Liu
2016 arXiv   pre-print
First we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015  ...  The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes.  ...  The aforementioned works lead us to the study of lower bounds for the minimum pair distance of constacyclic codes and constructions of MDS symbol-pair codes.  ...

### Improved Singleton bound on insertion-deletion codes and optimal constructions [article]

Bocong Chen, Guanghui Zhang
2021 arXiv   pre-print
bound were also constructed explicitly.  ...  Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any [n,k] linear code over 𝔽_q satisfies d≤2n-2k+2.  ...  The aforementioned works lead us to the study of Singleton-type bounds for the minimum insdel distance of general linear codes and optimal constructions of such codes.  ...
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