diff --git a/src/fec.rs b/src/fec.rs
index f1a9e7cb1ebfaa80d0c81194f34e7dc989b50392..a1fd2da61b99e39138be447af095367b4bdfa34c 100644
--- a/src/fec.rs
+++ b/src/fec.rs
@@ -209,6 +209,9 @@ mod tests {
 
     use super::recode_with_coeffs;
 
+    type LC = Vec<usize>;
+    type LCExclusion = Vec<usize>;
+
     fn bytes() -> Vec<u8> {
         include_bytes!("../tests/dragoon_32x32.png").to_vec()
     }
@@ -240,16 +243,20 @@ mod tests {
             &[3, 6, 8, 9, 10],
             &[vec![2, 4, 6], vec![1, 3, 7], vec![6, 7, 8], vec![3, 6, 8]],
         ));
+        assert!(contains_one_of(&[0, 4, 5], &[vec![2, 3, 5], vec![4, 5]]));
     }
 
     fn try_all_decoding_combinations<F: PrimeField>(
         data: &[u8],
         shards: &[Shard<F>],
         k: usize,
+        n: usize,
         test_case: &str,
         limit: Option<usize>,
-        should_not_be_decodable: Vec<Vec<usize>>,
+        should_not_be_decodable: Vec<LCExclusion>,
     ) {
+        let there_are_recoded_shards = shards.len() > n;
+
         for c in shards
             .iter()
             .cloned()
@@ -257,27 +264,52 @@ mod tests {
             .combinations(k)
             .take(limit.unwrap_or(usize::MAX))
         {
-            let s = c.iter().map(|(_, s)| s).cloned().collect();
-            let is: Vec<usize> = c.iter().map(|(i, _)| i).cloned().collect();
+            let is: Vec<usize> = c.iter().map(|(i, _)| *i).collect();
+            if there_are_recoded_shards {
+                let contains_recoded_shards = *is.iter().max().unwrap() < n;
+                if contains_recoded_shards {
+                    continue;
+                }
+            }
 
-            let actual = decode::<F>(s);
+            let pretty_is = is
+                .iter()
+                .map(|&i| {
+                    #[allow(clippy::comparison_chain)]
+                    if i == n {
+                        "(n)".into()
+                    } else if i > n {
+                        format!("(n + {})", i - n)
+                    } else {
+                        format!("{}", i)
+                    }
+                })
+                .collect::<Vec<_>>()
+                .join(", ");
+            let pretty_is = format!("[{pretty_is}]");
+
+            let actual = decode::<F>(c.iter().map(|(_, s)| s).cloned().collect());
 
             if contains_one_of(&is, &should_not_be_decodable) {
                 assert!(
                     actual.is_err(),
-                    "should not decode with {:?} {test_case}",
-                    is
+                    "should not decode with {} {test_case}",
+                    pretty_is
                 );
                 continue;
             }
 
-            assert!(actual.is_ok(), "could not decode with {:?} {test_case}", is);
+            assert!(
+                actual.is_ok(),
+                "could not decode with {} {test_case}",
+                pretty_is
+            );
 
             assert_eq!(
                 data,
                 actual.unwrap(),
-                "bad decoded data with {:?} {test_case}",
-                is,
+                "bad decoded data with {} {test_case}",
+                pretty_is,
             );
         }
     }
@@ -289,37 +321,29 @@ mod tests {
         let shards = encode::<F>(data, &Matrix::random(k, n, &mut rng))
             .unwrap_or_else(|_| panic!("could not encode {test_case}"));
 
-        try_all_decoding_combinations(data, &shards, k, &test_case, None, vec![]);
+        try_all_decoding_combinations(data, &shards, k, n, &test_case, None, vec![]);
     }
 
-    fn end_to_end_with_recoding_template<F: PrimeField>(data: &[u8], k: usize, n: usize) {
-        assert!(n >= 5, "n should be at least 5, found {}", n);
-
+    fn end_to_end_with_recoding_template<F: PrimeField>(
+        data: &[u8],
+        k: usize,
+        n: usize,
+        recoding_steps: Vec<LC>,
+        should_not_be_decodable: Vec<LCExclusion>,
+        name: &str,
+    ) {
         let mut rng = ark_std::test_rng();
-        let test_case = format!("TEST | data: {} bytes, k: {}, n: {}", data.len(), k, n);
+        let test_case = format!(
+            "TEST | data: {} bytes, k: {}, n: {}, scenario: {}",
+            data.len(),
+            k,
+            n,
+            name
+        );
 
         let mut shards = encode::<F>(data, &Matrix::random(k, n, &mut rng))
             .unwrap_or_else(|_| panic!("could not encode {test_case}"));
 
-        let recoding_steps = [
-            vec![2, 4],       // =  n
-            vec![1, 3],       // = (n + 1)
-            vec![n, (n + 1)], // = (n + 2) = ((2, 4), (1, 3))
-            vec![0],          // = (n + 3) = (0)
-            vec![(n + 3)],    // = (n + 4) = (0)
-        ];
-        let should_not_be_decodable = vec![
-            vec![2, 4, n],
-            vec![1, 3, (n + 1)],
-            vec![n, (n + 1), (n + 2)],
-            vec![1, 3, n, (n + 2)],
-            vec![2, 4, (n + 1), (n + 2)],
-            vec![1, 2, 3, 4, (n + 2)],
-            vec![0, (n + 3)],
-            vec![0, (n + 4)],
-            vec![(n + 3), (n + 4)],
-        ];
-
         for step in recoding_steps {
             let shards_to_recode: Vec<_> = shards
                 .iter()
@@ -330,18 +354,26 @@ mod tests {
             shards.push(recode_random(&shards_to_recode, &mut rng).unwrap().unwrap());
         }
 
-        try_all_decoding_combinations(data, &shards, k, &test_case, None, should_not_be_decodable);
+        try_all_decoding_combinations(
+            data,
+            &shards,
+            k,
+            n,
+            &test_case,
+            None,
+            should_not_be_decodable,
+        );
     }
 
     #[test]
     fn end_to_end() {
         let bytes = bytes();
 
-        for k in [3, 5] {
-            for rho in [0.5, 0.33] {
-                let n = (k as f64 / rho) as usize;
-                end_to_end_template::<Fr>(&bytes, k, n);
-            }
+        let ks = [3, 5];
+        let n = 5;
+
+        for k in ks {
+            end_to_end_template::<Fr>(&bytes, k, n);
         }
     }
 
@@ -349,10 +381,62 @@ mod tests {
     fn end_to_end_with_recoding() {
         let bytes = bytes();
 
-        for k in [3, 5] {
-            for rho in [0.50, 0.33] {
-                let n = (k as f64 / rho) as usize;
-                end_to_end_with_recoding_template::<Fr>(&bytes, k, n);
+        fn get_scenarii(n: usize) -> Vec<(String, Vec<LC>, Vec<LCExclusion>)> {
+            vec![
+                // ```mermaid
+                // graph TD;
+                //     a[n+1]; b[n+2]; c[n+3];
+                //
+                //     1;
+                //     3-->a; 5-->a;
+                //     2-->b; 4-->b;
+                //     a-->c; b-->c;
+                // ```
+                (
+                    "simple".into(),
+                    vec![
+                        vec![2, 4],       // =  n
+                        vec![1, 3],       // = (n + 1)
+                        vec![n, (n + 1)], // = (n + 2) = ((2, 4), (1, 3))
+                    ],
+                    vec![
+                        vec![2, 4, n],
+                        //
+                        vec![1, 3, (n + 1)],
+                        vec![n, (n + 1), (n + 2)],
+                        vec![1, 3, n, (n + 2)],
+                        vec![2, 4, (n + 1), (n + 2)],
+                        vec![1, 2, 3, 4, (n + 2)],
+                    ],
+                ),
+                // ```mermaid
+                // graph TD;
+                //     a[n+1]; b[n+2];
+                //
+                //     1-->a; a-->b;
+                //     2; 3; 4; 5;
+                // ```
+                (
+                    "chain".into(),
+                    vec![
+                        vec![0],   // = (n) = (0)
+                        vec![(n)], // = (n + 1) = (0)
+                    ],
+                    vec![vec![0, (n)], vec![0, (n + 1)], vec![(n), (n + 1)]],
+                ),
+            ]
+        }
+
+        for (k, n) in [(3, 5), (5, 5), (8, 10)] {
+            for (name, steps, should_not_decode) in get_scenarii(n) {
+                end_to_end_with_recoding_template::<Fr>(
+                    &bytes,
+                    k,
+                    n,
+                    steps,
+                    should_not_decode,
+                    &name,
+                );
             }
         }
     }