On two-weight 
                $$\mathbb {Z}_{2^k}$$
                
                    
                        
                            Z
                            
                                2
                                k
                            
                        
                    
                
            -codes

Shi, Minjia; Sepasdar, Zahra; Alahmadi, Adel; Solé, Patrick

doi:10.1007/s10623-017-0390-0

Cited by 29 publications

(14 citation statements)

References 12 publications

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“…13 Moreover, proteins unfold and refold spontaneously in vivo, 14 and the ability to fold with fast kinetics 15 and avoid metastable kinetic traps 16 is thought to be facilitated by properties of the denatured state that are not fully understood, 8 such as conformational preferences that preorganize an unfolded chain. 17 Despite these insights and the long-held recognition that a molecular-level understanding of folding and function requires the energy (or Boltzmann) weighted contributions of the myriad of structural states accessible to the protein, a quantitative thermodynamic description of the conformations in the denatured state has proven problematic.…”

Section: Introductionmentioning

confidence: 99%

“…18 Also, in the absence of folding, conformational selection is simplified and locally driven by factors such as hydration 19 and steric hindrance. 20 Peptides are found to have strong preferences for the left-handed polyproline II (PPII) backbone conformation, [21][22][23] suggesting that PPII structures may dominate the ensemble of conformations in the denatured state. Denatured proteins, however, exhibit structural attributes that are not well-modeled by short peptides.…”

Section: Introductionmentioning

confidence: 99%

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Impact of Heat on Coil Hydrodynamic Size Yields the Energetics of Denatured State Conformational Bias

et al. 2019

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Conformational equilibria in the protein denatured state have key roles regulating folding, stability, and function. The extent of conformational bias in the protein denatured state under folding conditions, however, has thus far proven elusive to quantify, particularly with regard to its sequence dependence and energetic character. To better understand the structural preferences of the denatured state, we analyzed both the sequence dependence to the mean hydrodynamic size of disordered proteins in water and the impact of heat on the coil dimensions, showing that the sequence dependence and thermodynamic energies associated with intrinsic biases for the α and polyproline II (PPII) backbone conformations can be obtained. Experiments that evaluate how the hydrodynamic size changes with compositional changes in the protein reveal amino acid specific preferences for PPII that are in good quantitative agreement with calorimetry-measured values from unfolded peptides and those inferred by survey of the protein coil library. At temperatures above 25 °C, the denatured state follows the predictions of a PPII-dominant ensemble. Heat effects on coil hydrodynamic size indicate the α bias is comparable to the PPII bias at cold temperatures.Though historically thought to give poor resolution to structural details, the hydrodynamic size of the unfolded state is found to be an effective reporter on the extent of the biases for the α and PPII backbone conformations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Impact of Heat on Coil Hydrodynamic Size Yields the Energetics of Denatured State Conformational Bias

et al. 2019

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“…,0,1,0). Note that, when t = 3 and s = 2, both solutions are (1,2) and (2, 0). By Theorem 4.1, for all the above solutions, we obtain a linear code H t1,...,ts .…”

Section: Classification Of Z 2 S -Linear Hadamard Codesmentioning

confidence: 99%

“…In [14], Krotov studied Z 2 s -linear Hadamard codes and their dual codes by using different generalizations of the Gray map. Recently, in [1], considering Carlet's generalization of the Gray map, two-weight Z 2 s -linear codes are studied. Note that Z 2 s -linear Hadamard codes are in fact a particular case of these two-weight codes.…”

Section: Introductionmentioning

confidence: 99%

On the Kernel of $$\mathbb {Z}_{2^s}$$-Linear Hadamard Codes

Fernández-Córdoba

Vela

Villanueva

2017

Coding Theory and Applications

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The Z2s -additive codes are subgroups of Z n 2 s , and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2sadditive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z4-linear Hadamard codes. In this paper, the kernel of Z2s -linear Hadamard codes and its dimension are established for s > 2. Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to t = 11 for any s ≥ 2, by using also the rank and, in some cases, further computations.

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“…Hadamard codes can be constructed from Hadamard matrices [1,16]. [21]. The homogeneous weight of an element i ∈ Z 2 s , denoted by wt(i), is such that wt(0) = 0, wt(2 s−1 ) = 2 s−1 and wt(i) = 2 s−1 for all i ∈ Z 2 s \{0, 2 s−1 }.…”

Section: Introductionmentioning

confidence: 99%

Equivalences among Z2s-linear Hadamard codes

Fernández-Córdoba

Vela

Villanueva

2020

Discrete Mathematics

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The Z 2 s -additive codes are subgroups of Z n 2 s , and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z 2 s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z 2 s -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z 2 s -linear Hadamard codes of length 2 t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t = 11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s ∈ {2, 3}, the full classification of the Z 2 s -linear Hadamard codes of length 2 t is established by giving the exact number of such codes.

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On two-weight $$\mathbb {Z}_{2^k}$$ Z 2 k -codes

Cited by 29 publications

References 12 publications

Impact of Heat on Coil Hydrodynamic Size Yields the Energetics of Denatured State Conformational Bias

Impact of Heat on Coil Hydrodynamic Size Yields the Energetics of Denatured State Conformational Bias

On the Kernel of $$\mathbb {Z}_{2^s}$$-Linear Hadamard Codes

Equivalences among Z2s-linear Hadamard codes

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