Decentralized exchange platforms such as Uniswap and Balancer operate on several pools where each pool contains two or more cryptocurrencies and constitutes direct trading pairs. The drawbacks here are that liquidity providing requires contribution of tokens in a specific proportion, and trading may require hopping between pools, hence increasing transaction fee and gas fee. We propose an automated market maker (AMM) protocol where liquidity providers can deposit any amount of tokens into the pool. The protocol will preserve the proportion of tokens by total value at the time of deposit and can be seen as a personalized self-balancing portfolio manager. In addition, since the invariant function is dynamic, all exchange pairs are executed from a single composite pool. Nevertheless, the scheme is vulnerable to flash loan attacks and must be used in conjunction with preventive measures.

Constructions of lattices are known to have a strong connection to the study of classical linear codes over F2; one such celebrated construction is that of Barnes-Wall (BW) lattices over ℤ[i], with i = √-1, wherein weighted sum of nested Reed-Muller codes over powers of the base 1 + i leads to the famous multi-level construction of BW lattices. Although these constructions facilitate simple encoding and decoding of information bits, the resulting codewords need to be mapped onto their representatives in order to reduce the average energy of the lattice code. Drawing inspirations from the case of BW lattices, in this work, we address a general question of how to arrive at representatives of the codewords of a lattice code over ℤ[θ], where θ is a quadratic integer, that is generated via a multi-level construction over linear codes over Fq, for q > 2. In particular, we introduce a novel shaping function τ : ℤ[θ] → ℤ[θ] on the components of lattice codewords, and prove that the natural constellation of lattices from multi-level constructions can be rearranged into a multi-dimensional cube or parallelepiped under such a map. We demonstrate numerically that our mapping results in a reduction of the average energy of lattice constellations. Our proposed mapping has applications in communications, particularly in encoding and decoding of lattice codes from multi-level constructions over q-ary linear codes.

We propose a lattice construction from totally real and complex multiplication fields, which naturally generalizes Construction A of lattices from p-ary codes obtained from the cyclotomic field Q(ζp), p a prime, which in turn contains the so-called Construction A of lattices from binary codes as a particular case. We focus on the maximal totally real subfield Q(ζpr+ ζp-r) of the cyclotomic field Q(ζpr), r ≥ 1. Our construction has applications to coset encoding of algebraic lattice codes for block fading channels, and in particular for block fading wiretap channels.

There is a rich theory of relations between lattices and linear codes over finite fields. However, this theory has been developed mostly with lattice codes for the Gaussian channel in mind. In particular, different versions of what is called Construction A have connected the Hamming distance of the linear code to the Euclidean structure of the lattice. This paper concentrates on developing a similar theory, but for fading channel coding instead. First, two versions of Construction A from number fields are given. These are then extended to division algebra lattices. Instead of the Euclidean distance, the Hamming distance of the finite codes is connected to the product distance of the resulting lattices, that is the minimum product distance and the minimum determinant respectively.

This paper considers pseudocodewords of lowdensity parity-check codes over alphabets with prime cardinality p for use over the p-ary symmetric channel. Pseudocodewords are decoding algorithm outputs that may not be legitimate codewords. Here, we consider pseudocodewords arising from graph cover decoding and linear programming decoding. For codes over the binary alphabet, such pseudocodewords correspond to rational points of the fundamental polytope. They can be characterized via the fundamental cone, which is the conic hull of the fundamental polytope; the pseudocodewords are precisely those integer vectors within the fundamental cone that reduce modulo 2 to a codeword. In this paper, we determine a set of conditions that pseudocodewords of codes over Fp, the finite field of prime cardinality p, must satisfy. To do so, we introduce a class of critical multisets and a mapping, which associates a real number to each pseudocodeword over Fp. The real numbers associated with pseudocodewords are subject to lower bounds imposed by the critical multisets. The inequalities are given in terms of the parity-check matrix entries and critical multisets. This gives a necessary and sufficient condition for pseudocodewords of codes over F2and F3and a necessary condition for those over larger alphabets. In addition, irreducible pseudocodewords of codes over F3are found as a Hilbert basis for the lifted fundamental cone.

We consider the problem of communication over a block fading wiretap channel. It is known that coding for such a channel can be done using nested lattice codes constructed over totally real number fields. In this paper, we propose a method for encoding an integral lattice over the ring of integers of a totally real number field, and study in particular the case of Q(ζp+ζp-1) using a linear code over Fp. This generalizes the well-known Construction A and provides an efficient coset encoding for algebraic lattices.

The performance of message-passing iterative decoding and linear programming decoding depends on the Tanner graph representation of the code. If the underlying graph contains cycles, then such algorithms could produce a noncodeword output. The study of pseudocodewords aims to explain this noncodeword output. We examine the structure of the pseudocodewords and show that there is a one-to-one correspondence between graph cover pseudocodewords and integer points in a lifted fundamental cone. This gives a simple proof that the generating function of the pseudocodewords for a general parity-check code is rational (a fact first proved by Li, Lu, and Wang (Lecture Notes in Computer Science, vol. 5557, 2009) via other methods). Our approach yields algorithms for producing this generating function and provides tools for studying the irreducible pseudocodewords. Specifically, Barvinok's algorithm and the Barvinok-Woods projection algorithm are applied, and irreducible pseudocodewords are found via a Hilbert basis for the lifted fundamental cone.