Families of new, multi-level integer 2D arrays are introduced here as an extension of the well-known binary Legendre sequences that are derived from quadratic residues. We present a construction, based on Fourier and Finite Radon Transforms, for families of periodic perfect arrays, each of size \(p\times p\) for many prime values p. Previously delta functions were used as the discrete projections which, when back-projected, build 2D perfect arrays. Here we employ perfect sequences as the discrete projected views. The base family size is \(p+1\). All members of these multi-level array families have perfect autocorrelation and constant, minimal cross-correlation. Proofs are given for four useful and general properties of these new arrays. 1) They are comprised of odd integers, with values between at most \(-p\) and \(+p\), with a zero value at just one location. 2) They have the property of ‘conjugate’ spatial symmetry, where the value at location (i, j) is always the negative of the value at location \((p-i, p-j)\). 3) Any change in the value assigned to the array’s origin leaves all of its off-peak autocorrelation values unchanged. 4) A family of \(p+1\), \(p\times p\) arrays can be compressed to size \((p+1)^2\) and each family member can be exactly and rapidly unpacked in a single \(p\times p\) decompression pass.

这里引入了新的多级整数二维数组系列，作为从二次留数导出的众所周知的二进制勒让德序列的扩展。我们提出了一种基于傅立叶和有限拉东变换的周期性完美数组族的构造，每个数组的大小为\(p\times p\)对于许多素值p。以前，Delta 函数被用作离散投影，当进行反投影时，可以构建二维完美数组。在这里，我们采用完美序列作为离散投影视图。基本族大小为\(p+1\)。这些多级阵列系列的所有成员都具有完美的自相关性和恒定、最小的互相关性。给出了这些新数组的四个有用且通用的属性的证明。1) 它们由奇数整数组成，值最多在\(-p\)和\(+p\)之间，并且仅在一个位置有零值。2) 它们具有“共轭”空间对称性，其中位置 ( i ,  j ) 处的值始终是位置\((pi, pj)\)处值的负数。3) 分配给数组原点的值的任何更改都会使其所有非峰值自相关值保持不变。4) 一个由\(p+1\) , \(p\times p\)数组组成的族可以被压缩到大小\((p+1)^2\)并且每个族成员都可以被精确且快速地解包在一个单次\(p\times p\)减压过程。