With the recent dramatic increase in performance of artificial intelligence and related computing systems, together with advanced sensing, connectivity, and technological platforms, autonomous systems are poised to enter many application domains such as transportation and manufacturing. As autonomy increases, the risks of cybersecurity threats are equally rising, requiring the development of sophisticated methods on all layers of autonomous systems architectures. Therefore, this paper systematically introduces cybersecurity challenges ranging from the physical layer to the system of systems layer defining the collaboration of autonomous systems. Without loss of generality, autonomous vehicles are used to highlight current developments, illustrating which efforts are necessary to achieve secure and safe autonomous systems. Our discussions are comprehensively highlighting which research domains require further investigation and offer promising opportunities to contribute to mitigating cybersecurity challenges of autonomous systems.

Generalized Goppa codes are defined by a code locator set$L$of polynomials and a Goppa polynomial G(x). When the degree of all code locator polynomials in$L$is one, generalized Goppa codes are classical Goppa codes. In this work, binary generalized Goppa codes are investigated. First, a parity-check matrix for these codes with code locators of any degree is derived. A careful selection of the code locators leads to a lower bound on the minimum Hamming distance of generalized Goppa codes which improves upon previously known bounds. A quadratic-time decoding algorithm is presented which can decode errors up to half of the minimum distance. Interleaved generalized Goppa codes are introduced and a joint decoding algorithm is presented which can decode errors beyond half the minimum distance with high probability. Finally, some code parameters and how they apply to the Classic McEliece post-quantum cryptosystem are shown.

A method is provided for constructing upper triangular square matrices over the univariate polynomial ring over a finite field, under certain constraints on the eigenvalues of the matrices. In some cases of interest, the degree of the determinant of such matrices is shown to be the smallest possible. The method is then applied to construct generator polynomial matrices of quasi-cyclic codes for correcting phased burst errors. Finally, an interpolation-based list decoding algorithm is presented for these codes, which, for a wide range of code parameters, is shown to outperform existing list decoding schemes.

An explicit construction of an infinite family of cyclic codes is presented which, over GF(4) (resp., GF(8)), have approximately 8/9 (resp., 48/49) the redundancy of BCH codes of the same minimum distance and length. As such, the new codes are the best codes currently known in a regime where the minimum distance is fixed and the code length goes to infinity.

An explicit construction of an infinite family of cyclic codes is presented which, over GF(4) (resp., GF(8)), have approximately 8/9 (resp., 48/49) the redundancy of BCH codes of the same minimum distance and length. As such, the new codes are the best codes currently known in a regime where the minimum distance is fixed and the code length goes to infinity.

A method is provided for constructing upper-triangular square matrices over the univariate polynomial ring over a finite field, under certain constraints on the eigenvalues of the matrices. In some cases of interest, the degree of the determinant of such matrices is shown to be the smallest possible. The method is then applied to construct generator polynomial matrices of quasi-cyclic codes with a prescribed designed minimum distance.

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row- and the column-code. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code.

New optimal constructions of locally repairable codes over small fields and their polynomial-time erasure list decoding are considered. Our code constructions are based on generalized code concatenation and give optimal binary codes with locality r = 2; 3. The impact of alphabet-dependent list recovery for alternant codes when applied to erasure list decoding of our constructed binary locally repairable codes is analyzed.

This paper studies bounds and constructions of locally repairable codes (LRCs) with multiple localities so-called multiple-locality LRCs (ML-LRCs). In the simplest case of two localities some code symbols of an ML-LRC have a certain locality while the remaining code symbols have another one. We extend two bounds, the Singleton and the alphabet-dependent upper bound on the dimension of Cadambe-Mazumdar for LRCs, to the case of ML-LRCs with more than two localities. Furthermore, we construct Singleton-optimal ML-LRCs codes.

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row and column codes. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm. More specifically, we prove an explicit (unreduced) basis of an ℓAℓB-quasi-cyclic product code in terms of the generator matrix in reduced Grobner basis with respect to the position-over-term (RGB/POT) order form of the ℓA-quasi-cyclic row code and the ℓB-quasicyclic column code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where ℓA= ℓB= 1). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an ℓAℓB-quasi-cyclic product code for two special cases: i) for ℓA= 2 and ℓB= 1 and ii) if the row code is a one-level ℓA-quasi-cyclic code (for arbitrary ℓA) and ℓB= 1. For arbitrary ℓAand ℓB, the PreRGB/POT form of the generator matrix of an ℓAℓB-quasi-cyclic product code is conjectured. The spectral analysis is applied to the generator matrix of the product of an ℓ-quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given ℓ-quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for ℓ-phased burst errors with guaranteed decoding radius.

This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound.

An interpolation-based list decoding algorithm for ℓ-quasi-cyclic codes over finite fields is developed and its guaranteed decoding radius for ℓ-phased burst errors is proven. It is also shown that for this error model and for certain parameter ranges, this new approach is advantageous over existing schemes.

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed- Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.

A new lower bound on the minimum Hamming distance of linear quasi-cyclic codes over finite fields is proposed. It is based on spectral analysis and generalizes the Semenov-Trifonov bound in a similar way as the Hartmann-Tzeng bound extends the BCH approach for cyclic codes. Furthermore, a syndrome-based algebraic decoding algorithm is given.

Two generalizations of the Hartmann-Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved. Our technique of embedding the code into a cyclic product code can be applied to other bounds, too and therefore generalizes them.

A new approach to bound the minimum distance of q-ary cyclic codes is presented. The connection to the BCH and the Hartmann-Tzeng bound is formulated and it is shown that for several cases an improvement is achieved. We associate a second cyclic code to the original one and bound its minimum distance in terms of parameters of the associated code.

A new lower bound on the minimum distance ofq-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem bound and, for some codes, upon the Hartmann-Tzeng bound. Several Boston bounds are special cases of our bound. For some classes of codes, the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.

A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum distance is proven and several classes of binary cyclic codes are identified. For some classes of codes, this bound is better than the known bounds (e.g. BCH or Hartmann-Tzeng bound). Furthermore, a quadratic-time decoding algorithm up to this new bound is developed.

The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an (n,k) Reed-Solomon code, a multiplicitysand a list sizel, our algorithm has time complexityO(ls4n2).

The Welch-Berlekamp approach for Reed-Solomon (RS) codes forms a bridge between classical syndrome-based decoding algorithms and interpolation-based list-decoding procedures for list size ℓ = 1. It returns the univariate error-locator polynomial and the evaluation polynomial of the RS code as a y-root. In this paper, we show the connection between the Welch-Berlekamp approach for a specific Interleaved Reed-Solomon code scheme and the Guruswami-Sudan principle. It turns out that the decoding of Interleaved RS codes can be formulated as a modified Guruswami-Sudan problem with a specific multiplicity assignment. We show that our new approach results in the same solution space as the Welch-Berlekamp scheme. Furthermore, we prove some important properties.

We modify the Euclidean algorithm of Feng and Tzeng to decode Reed-Solomon (RS) codes up to the Sudan radius. The basic steps are the virtual extension to an Interleaved RS code and the reformulation of the multi-sequence shift-register problem of varying length to a multi-sequence problem of equal length. We prove the reformulation and analyze the complexity of our new decoding approach. Furthermore, the extended key equation, that describes the multi-sequence problem, is derived in an alternative polynomial way.

In 2000 Roth and Ruckenstein proposed an extended key equation for solving the interpolation step in the Sudan decoding algorithm. Generalizing their idea, a sequence of key equations for the Guruswami-Sudan (GS) algorithm, which is able to list decode a Reed-Solomon code with arbitrary rate, is derived. This extension allows a reduction of the number of equations and therefore a reduction of the algorithmpsilas complexity. Furthermore, we indicate how to adapt the fundamental iterative algorithm for block Hankel matrices and thus solving the GS-interpolation step efficiently.