In other words:



> 
> What is $\mathrm{Ext}\_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?
> 
> 
> 


If the 4-line is not known, how much is known about it?


Here, $\mathcal{A}$ is the 2-primary Steenrod algebra, $4$ is the homological degree corresponding to the Adams filtration, and $t$ is the internal grading degree. Those $\mathrm{Ext}$ groups make up the fourth row of the classical Adams spectral sequence $E\_2 = \mathrm{Ext}\_{\mathcal{A}}^{s,t}(\mathbb{Z}/2,\mathbb{Z}/2)$ converging to the 2-adic completion of the $(t-s)^{\mathrm{th}}$ stable homotopy group of the sphere.


For context, 


* the 1-line is generated by the classes $h\_i$, $i \geq 0$, ($\mathrm{deg}\: h\_i = (1,2^i)$),
* the 2-line is generated by the product classes $h\_i h\_j$, subject to the relations $h\_i h\_{i+1} = 0$ and $h\_i h\_j = h\_j h\_i$,
* the 3-line is generated by two sets of classes, 


	1. the product classes $h\_i h\_j h\_k$, subject to the relations implied by $h\_i h\_{i+2}^2 = 0$, $h\_{i+1}^3 = h\_i^2 h\_{i+2}$, $h\_i h\_{i+1} = 0$, and $h\_i h\_j = h\_j h\_i$,
	2. the Massey products $\langle h\_{i+1},h\_i,h\_{i+2}^2 \rangle$.



Is the 4-line of the E_2 term of the classical Adams spectral sequence known?
The 4-line is determined by Wen-Hsiung Lin in "$Ext\_A^{4,\*}({\bf Z}/2,{\bf Z}/2) $ and $Ext\_A^{5,\*}({\bf Z}/2,{\bf Z}/2) $", Topology and its Applications (2008) vol 155 no.5 pp 459-496.


He gives a basis for the indecomposable elements in $Ext\_A^{4,\*}$ and generators and relations for the quotient of $Ext\_A^{s,\*} $ for $s \le 4$ by the indecomposables of $Ext\_A^{4,\*}


